The Pennsylvania State University

The Graduate School

Department of Mathematics

RANKS OF PARTITIONS AND DURFEE SYMBOLS

A Thesis in

Mathematics

by

William J. Keith

c© 2007 William J. Keith

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Doctor of Philosophy

August 2007

ii

Committee Page

The thesis of William Keith has been reviewed and approved* by the following:

Dr. George Andrews

Professor, Mathematics

Thesis Adviser

Chair of Committee

Dr. Wen-Ching W. Li

Professor, Mathematics

Dr. Ae Ja Yee

Assistant Professor, Mathematics

Dr. Martin Furer

Professor, Computer Science

Dr. John Roe

Professor, Mathematics

Department Head, Mathematics

*Signatures are on file in the Graduate School.

iii

Abstract

This thesis presents generalizations of several partition identities related to the

rank statistic.

One set of these is new: k-marked Durfee symbols, as defined in a paper by An-

drews. This thesis extends and elaborates upon several congruence theorems presented

in the paper that originated those objects, showing that an infinite family of such theo-

rems exists. The number of l-marked Durfee symbols of n are related to the distribution

of ranks of partitions of n modulo 2l+ 1; the relationship is made explicit and explored

in various directions.

Another set of identities deals with the very classical theorem of Euler on par-

titions into odd and distinct parts. This was given bijective proof by Sylvester, giving

occasion to discover new statistical equalities, which in turn were generalized to parti-

tions into parts all ≡ c (modm) by Pak, Postnikov, Zeng, and others. This work further

extends the previous theorems to partitions with residues (modm) that differ but do not

change direction of difference, i.e. residues monotonically rise or fall.

Attached as an appendix is a translation of the thesis of Dieter Stockhofe, Bijektive

Abbildungen auf der Menge der Partitionen einer naturlichen Zahl. This is provided in

support of the tools therefrom used in Chapter 3, as well as in the spirit of a service to

the Anglophone mathematical community.

iv

Table of Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Durfee Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Fine’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 2. The Full Rank: Congruences and Complete Behavior . . . . . . . . . 9

2.1 Prime Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Nonprime Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Chapter 3. Generalizing Sylvester’s Bijection . . . . . . . . . . . . . . . . . . . . 27

3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Sylvester’s Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Generating Functions . . . . . . . . . . . . . . . . . . . . . . 48

3.2.2 Appearance of Descents . . . . . . . . . . . . . . . . . . . . . 53

Appendix A. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Appendix B. Translation: Stockhofe’s Thesis . . . . . . . . . . . . . . . . . . . . 63

B.1 Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

B.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

v

B.3 The q-Modular Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 67

B.4 Construction of Lq . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

B.4.1 q-flat Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . 77

B.4.2 A Smaller Bijective Transformation . . . . . . . . . . . . . . . 81

B.4.3 A Larger Transformation . . . . . . . . . . . . . . . . . . . . 86

B.4.4 Generalizing Conjugation . . . . . . . . . . . . . . . . . . . . 89

B.4.5 The Bijection Lq . . . . . . . . . . . . . . . . . . . . . . . . . 93

B.5 Some Counting Theorems . . . . . . . . . . . . . . . . . . . . . . . . 100

B.6 The Special Case q = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 105

B.7 The Fixed Points of Lq . . . . . . . . . . . . . . . . . . . . . . . . . . 114

B.8 Groups of Permutations of P (n) . . . . . . . . . . . . . . . . . . . . . 117

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

vi

List of Figures

1.1 The Ferrers diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The m-modular diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Illustration of O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Illustration of η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 The two-residue case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Cases of O with descents in ρ. . . . . . . . . . . . . . . . . . . . . . . . . 56

B.1 Part (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

B.2 Part (ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

B.3 A column inserted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

B.4 An angle inserted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

B.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

B.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

B.7 A sketch of the process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

B.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

B.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

B.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

B.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

vii

B.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

B.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

viii

Acknowledgments

I am foremost indebted to my advisor, Dr. George Andrews, for the mathematical

mentoring he has provided and for generous support through a long graduate career.

1

Chapter 1

Introduction

In this chapter we lay out the basics of the theory of partitions and provide at

least the minimal definitions and toolkit necessary for any reader to understand what

this thesis is about and how it relates to previous work.

We say an integer vector λ = (λ1, . . . λk) is a partition of n if λ1 ≥ · · · ≥ λk ≥ 1

and λ1 + · · ·+ λk = n. The number of partitions of n shall be denoted p(n). A common

method for illustrating a partition, especially when we want to construct bijections be-

tween sets of partitions of various types, is the Ferrers diagram, consisting of a lattice of

dots, each column representing the i-th part and being of height λi:

Fig. 1.1. The Ferrers diagram and Durfee square of λ = (8, 8, 7, 5, 5, 5, 5, 4, 3, 2, 2, 1).

An example of a bijection on partitions immediately suggested by the Ferrers

diagram is conjugation, in which we map a partition λ to the partition φ illustrated by

2

the diagram of λ transposed about its main diagonal. This common and fundamental

procedure we will label throughout this thesis φ = λ′.

It is clear that conjugation preserves the length of the main diagonal and the size

of the largest square that can fit in a partition’s diagram from the upper left corner,

indicated on the illustration above. This square is called the Durfee square of the par-

tition. Non-diagrammatically, we can state that the size c of the Durfee square in λ is

the largest c such that λc − c ≥ 0. In the above illustration, c = 5.

In his 1944 paper ”Some Guesses in the Theory of Partitions,” Freeman Dyson

introduced in 5 pages flat a tool of enormous utility to the field: his rank statistic for

partitions, defined quite simply as the largest part of a partition, minus the number of

parts. Letting N(m,n) be the number of partitions of n with rank m, the generating

function of this statistic is

R1(z; q) =∞∑

m=−∞

∑n≥0

N(m,n)zmqn =∑n≥0

qn2

(zq; q)n(q/z; q)n(1.1)

where (a; q)n =∏n−1i=1 (1 − aqi). Common methods in combinatorial theory interpret

the right hand side of the equation above as a sum over Ferrers diagrams with Durfee

squares of size n (a square starts with rank 0), and with rows of length no more than n

below the Durfee square (contributing z each) and columns of length no more than n to

the right of the Durfee square (contributing z−1). It is easily observed that conjugation

negates rank, and so N(m,n) = N(−m,n), a fact of which we make use in Chapter 2.

Originally, Dyson’s introduction of this construction was motivated by the fact

that partitions with rank ≡ i (mod 5, 7) were distributed evenly for partitions of 5n+ 4

3

and 7n+ 5 respectively: that is, if N(i, p, n) denotes the number of partitions of n with

rank ≡ i (mod p), then N(i, 5, 5n+4) = N(j, 5, 5n+4) and N(i, 7, 7n+5) = N(j, 7, 7n+5)

for all i, j. This provided a combinatorial explanation of Ramanujan’s famous theorems

that p(5n+4) ≡ 0 (mod 5) and p(7n+5) ≡ 0 (mod 7). (Though it failed to prove the third

theorem, that p(11n+ 6) ≡ 0 (mod 11); this theorem, and the many related congruences

later produced first by Ramanujan and collaborators, and in more recent years by Ken

Ono and his school, awaited a related statistic called the crank, which Dyson conjectured

but was unable to find. This was done by Frank Garvan, in partial concert with George

Andrews, decades later.) Since then the rank has developed in rich and unexpected

directions, two of which are studied in this thesis.

1.1 Durfee Symbols

In studying further partition congruences, A.O.L. Atkin and Frank Garvan [2]

related the rank and the crank via a differential equation, in doing so constructing the

k-th moments of the rank function. George Andrews [1] has in turn constructed the

symmetrized k-th moment

ηk(n) =∞∑

m=−∞

(m+ bk−1

2 ck

)N(m,n)

and associated to these objects the k-marked Durfee symbol, in which a partition is

decomposed as described in Equation 1.1 and the columns and rows about the Durfee

square are marked with k subscripts or colors, according to the following rules:

4

Definition 1. The ordered, subscripted vector pair

t1 t2 . . . tr

b1 b2 . . . bs

c

is a k-marked

Durfee symbol of n = c2 + t1 + . . . tr + b1 + · · ·+ bs if

• ti, bj∈ 11, 12, . . . , 1k, 21, 22, . . . , 2k, . . . , c1, . . . , ck;

• i > j, ti(resp. b

i) = a

b, tj(resp. b

j) = d

e⇒ a ≥ d, b ≥ e;

• Every subscript 1, . . . , k − 1 appears at least once in the top row;

• If M1,M2, . . .Mk−2,Mk−1 are the largest parts with their respective subscripts in

the top row, then bi= d

e⇒ d ∈ [M

e−1,Me], setting M1 = 1 and M

k= c.

If we then call Dk(n) the number of k-marked Durfee symbols of n, then

Dk+1(n) = η2k(n) (Corollary 13 in [1]). The study of congruence theorems for Durfee

symbols thus informs the study of congruence theorems for partitions of standard type.

Making this information explicit requires defining a richer rank these objects bear

called the full rank, preserving some of the properties of the k-coloration:

Definition 2. Let δ be a k-marked Durfee symbol and let τi(resp. β

i) be the number

of parts in the top (resp. bottom) row with subscript i. Then the ith-rank of a Durfee

symbol is

ρi(δ) =

τi− β

i− 1 1 ≤ i < k

τi− β

ii = k

.

Definition 3. The full rank of a k-marked Durfee symbol δ is ρ1(δ)+2ρ2(δ)+3ρ3(δ)+

· · ·+ kρk(δ).

5

We set Dk(m1, . . . ,mk;n) to be the number of k-marked Durfee symbols with ith

ranks all mi. In analogy to our previous construction for the rank we call NF

l(m,n) the

number of l-marked Durfee symbols of n with full rank m, and NFl(b, p, n) the number

of l-marked Durfee symbols of n with full rank ≡ b (mod p).

Andrews produces the generating function (Theorems 10 and 7 in [1]):

∞∑n1,...,nk=−∞

∑n≥0

Dk(n1, . . . , nk;n)x1

n1 . . . xknkq

n = Rk(x1, . . . xk;n)

=k∑i=1

R1(xi; q)

k∏j=1j 6=i

(xi− x

j)(1− x

i−1x

j−1)

. (1.2)

This theorem in hand, he produces two congruences: that D2(n) ≡ 0 (mod 5) for

n ≡ 1, 4 (mod 5) and D3(n) ≡ 0 (mod 7) for n ≡ 0, 1, 5 (mod 7), because NF2(i, 5, n) =

NF2(j, 5, n) and NF3(i, 7, n) = NF3(j, 7, n) for all i, j in those progressions. Further-

more, it transpires that for n 6≡ 1, 4 (mod 5) or n 6≡ 0, 1, 5 (mod 7), we nevertheless have

NFl(i, 5, n) = NF

l(j, 5, n) and NF

l(i, 7, n) = NF

l(j, 7, n) for all i, j 6= 0 in any progres-

sion.

It is our intent in the next chapter to put the above two theorems in a more

general setting. We show that they are the simplest two examples of an infinite family of

related theorems; we explore the failure mode of the latter cases and explain by exactly

how much they fail, giving rise to an infinite family of congruences for prime modulus;

6

and we examine to full detail the behavior of the residue classes for nonprime (odd)

modulus.

1.2 Fine’s Theorems

Chapter 3 of this thesis establishes identities that at once refine and generalize

the rank, combining lines pursued separately by previous authors, particularly Glaisher

and Fine.

In the inaugural work of partition theory, Chapter 16 of ”Introductio in Analysin

Infinitorum,” Leonhard Euler shows that the number of partitions of n into odd parts are

equinumerous with those in which parts are distinct. Glaisher later generalized this to

partitions into parts not divisible by m; N.J. Fine refines it by showing that the number

of partitions of n into odd parts, with largest part plus twice the number of parts equal

to 2M + 1, equals the number of partitions of n into distinct parts with largest part M .

The penultimate statistic here can be regarded as one instance of a generalization of the

rank, the (a, b)-rank: a times the largest part of a partition, minus b times the number

of parts. Here, we have the (1,−2)-rank. Fine’s proof is via generating functions; the

theorem can also be proven bijectively by a transformation of Sylvester, an m-modular

generalization of which becomes our primary tool to prove a similar theorem for the

(1,−m)-rank.

In exploring this a different diagrammatic presentation of partitions is useful:

the m-modular diagram. In this presentation, we fix a modulus m and display each

λi= k

im+ j

iby writing a column consisting of k

irepetitions of m, topped (or founded,

7

in which case we present the diagram marked by *) with ji, if j

iis nonzero (0 ≤ j

i< m).

Figure 1.2 provides an example.

Fig. 1.2. The 5-modular diagram of λ = (22, 19, 15, 13, 7, 6, 2).

It will be quickly observed that conjugation is no longer a simply-defined opera-

tion. An m-modular analogue of conjugation was produced in 1981 as the doctoral thesis

of one Dieter Stockhofe [13], in the process producing several useful tools we employ in

Chapter 3. Also, the notion of a Durfee square is necessarily somewhat coarser for an

m-modular diagram; denoted dm

(λ) (resp. dm∗(λ)), we can define its size as the largest

i such that bλim c ≥ i (resp. dλi

m e ≥ i).

On the other hand, when m > 1 (m = 1 gives us the original Ferrers diagram)

there are more interesting statistics regarding the parts than simply their number. We

can treat the list of nonzero residues (modm) of such parts appearing as a multiset in

[1, . . . ,m−1] and construct a more structured statistic taking into account combinatorial

statistics on: the number of nonzero residues, the number of kinds of residues appearing,

8

the number of descents in the list of residues read from left to right, and perhaps most

interestingly sequences of consecutive parts. The observer will note that when m = 2,

as in the case for odd partitions, the residues are all 1, and thus the number of kinds

of parts is always 1, and the number of descents is necessarily 0; this behavior hides a

degeneracy of these statistics which flowers to great effect in the case for higher modulus.

In Chapter 3 we examine the work of Sylvester, Pak, Postnikov, Zeng, and others

on (m, c) partitions, which are partitions into parts not divisible by m in which all parts

have the same residue c (modm). Their work is a step toward Glaisher’s generalization

of Euler’s theorem, but has the same degeneracies in the number of kinds of parts and

the number of descents as those discussed for the m = 2 case. We do not yet obtain

the full Glaisher-style generalization for the statistic involving sequences of parts, but

we can describe a more general theorem for m-falling or m-rising partitions, in which

residues of parts 6≡ 0(modm) ascend or descend monotonically.

9

Chapter 2

The Full Rank: Congruences and Complete Behavior

2.1 Prime Moduli

In the previous chapter, we observed two previously-proven theorems on the con-

gruence behavior of the full rank of 2-marked and 3-marked Durfee symbols, in arithmetic

progressions mod 5 and 7 respectively. These are specific instances of a general theorem

for the full ranks of k-marked Durfee symbols in arithmetic progressions of any odd

modulus:

Theorem 1. Let p = 2l + 1 ∈ Z, p ≥ 5. Say NFl(j, p, pn + d) is the number of

l-marked Durfee symbols of pn + d with full rank congruent to j mod p. Then, if

gcd(i, p) = gcd(j, p), we have NFl(i, p, pn+ d) = NF

l(j, p, pn+ d) .

The inverse does not hold. More generally, when p is not prime the investigation

of the differences between divisor-groups of residue classes is itself interesting. As a

corollary of this theorem, since all residues not congruent to zero are coprime to a prime

modulus we have the near-equidistribution

Corollary 1. If p = 2l + 1 is prime, p ≥ 5, then NFl(i, p, pn + d) = NF

l(j, p, pn + d)

for all i, j 6≡ 0 mod p.

10

This is the case for the two theorems previously discussed. The additional behav-

ior of complete equidistribution in residue classes comes about due to a second conse-

quence that will be easily seen from the theorem’s method of proof:

Theorem 2. If p = 2l + 1 is prime, p ≥ 5, then

NFl(0, p, pn+ d)−NF

l(1, p, pn+ d) = N(l − 1, p, pn+ d)−N(l, p, pn+ d).

Combining Corollary 1 and Theorem 2, then,

Corollary 2. If p = 2l + 1 is prime, p ≥ 5, then

∞∑m=−∞

NFl(m,n) = D

l(n) ≡ N(l − 1, p, n)−N(l, p, n) (mod p).

Because this difference is 0 for p = 5, d = 1, 4 and p = 7, d = 0, 1, 5, we have full

equidistribution and a clean congruence theorem in those progressions.

Proof of Theorem 1. Our basic strategy, as in [1], is to observe∑∞n=1

∑p−1

b=0NF

l(b, p, n)ζ

pbqn,

where ζp

is a primitive p-th root of unity. To prove the general theorem requires the

additional observation that this sum, in terms of the rank, behaves well with respect to

sums of conjugate powers of ζp. To make this precise, we break the sum down thus:

11

∞∑n=1

p−1∑b=0

NFl(b, p, n)ζ

pbqn = R

l(ζp, ζp2, . . . , ζ

pl; q)

=l∑

i=1

R1

(ζpi; q)

l∏j=1j 6=i

(ζpi − ζ

pj)(

1− ζp−i−j

)

=l∑

i=1

l∏j=1j 6=i

(ζpi − ζ

pj)(

1− ζp−i−j)

−1

·∞∑n=1

p−1∑k=0

p−1∑d=0

ζpikN(k, p, pn+ d)qpn+d

=l∑

i=1

l∏j=1j 6=i

(ζpi − ζ

pj)(

1− ζp−i−j)

−1

·p−1∑d=0

qd ∑n≥1

p−1∑k=0

ζpikN(k, p, pn+d)qpn .

(2.1)

Following Atkin, we define ra,b

(q; p; d) =∑n≥0 q

n(N(a, p, n)−N(b, p, n)). Then,

for any given d,

∑n≥1

N(l, p, pn+ d)qpn =∑n≥1N(l − 1, p, pn+ d)qpn − r

l−1,l(qp; p; d)

=∑n≥1N(l − 2, p, pn+ d)qpn − r

l−2,l(qp; p; d)

= . . .

=∑n≥1N(0, p, pn+ d)qpn − r0,l(q

p; p; d) .

12

We further note that the evenness of the rank generating function for partitions

(a classic example of a bijective proof: conjugate the partition) gives us the identities

N(l, p, pn+d) = N(p− l, p, pn+d) and thus rb,c

(qp; p; d) = rp−c,p−b(q

p; p; d). Combined

with the previous line and the fact that∑p−1

b=0ζpb = 0, we have

∑n≥1

N(0, p, pn+ d)qpn − r0,l(qp; p; d)

ζpi·0

+

∑n≥1

N(1, p, pn+ d)qpn − r1,l(qp; p; d)

ζpi·1 + . . .

+

∑n≥1

N(l, p, pn+ d)qpn ζ

pi·l +

∑n≥1

N(l + 1, p, pn+ d)qpn ζ

pi·(l+1)

+ · · ·+

∑n≥1

N(p− 1, p, pn+ d)qpn − r1,l(qp; p; d)

ζpi·(p−1) = 0 . (2.2)

(For use in a later theorem we note that it matters in the above calculation that

i 6≡ 0 mod p in this context, but its value otherwise is irrelevant; if p is nonprime and

gcd(i, p) 6= 1, we have merely employed the same identity pgcd(i,p) times.)

Thus, gathering the N(k, p, pn+ d) terms,

p−1∑k=0

ζpik ∑n≥1

N(k, p, pn+ d)qpn = r0,l(qp; p; d) +

l−1∑g=1

rg,l

(qp; p; d)(ζpig + ζ

pi(−g)

).

13

Thence

∞∑n=1

p−1∑b=0

NFl(b, p, n)ζ

pbqn =

l∑i=1

l∏j=1j 6=i

(ζpi − ζ

pj)(

1− ζp−i−j)

−1

·p−1∑d=0

qd

r0,l(qp; p; d) +l−1∑g=1

rg,l

(qp; p; d)(ζpig + ζ

pi(−g)

) . (2.3)

For any n, then, we have by equation of coefficients in powers of q that

p−1∑b=0

NFl(b, p, n)ζ

pb =

l∑i=1

l∏j=1j 6=i

(ζpi − ζ

pj)(

1− ζp−i−j)

−1

·

N(0, p, n)−N(l, p, n) +l−1∑g=1

(N(g, p, n)−N(l, p, n))(ζpig + ζ

pi(−g)

) . (2.4)

To prove the theorem, it suffices to show that the right-hand side of 2.4 is an

integer. The constant term that appears before the sum contributes only 0: notice that∏lj=1j 6=i

(ζpi − ζ

pj)(

1− ζp−i−j) = ζ

pi(l−1)

∏lj=1j 6=i

(1− ζ

p−i+j)(1− ζ

p−i−j),

and the exponents −i+ j,−i− j | 1 ≤ j ≤ l, j 6= i are precisely 1, . . . , p−1\0, i, 2i

when reduced mod p. Since∏p−1

i=1

(1− ζ

pi)

= p, we can simplify the term thus:

14

(N(0, p, n)−N(l, p, n)) ·l∑

i=1

l∏j=1j 6=i

(ζpi − ζ

pj)(

1− ζp−i−j)

−1

= (N(0, p, n)−N(l, p, n)) · 1p·l∑

i=1ζp−i(l−1) (1− ζ

p−2i)(1− ζ

p−i)

= (N(0, p, n)−N(l, p, n)) · 1p·l∑

i=1

(ζp−i(l−1) + ζ

p−i(l+2) − ζ

p−i(l+1) − ζ

p−i(l)

)

= (N(0, p, n)−N(l, p, n)) · 1p·l∑

i=1

(ζp−i(l−1) + ζ

pi(l−1) − ζ

p−i(l+1) − ζ

pi(l+1)

)

= (N(0, p, n)−N(l, p, n)) · 1p·

2l∑i=1

(ζpi(l−1) − ζ

p−i(l+1)

)

= (N(0, p, n)−N(l, p, n)) · 1p· (−1− (−1)) = 0 . (2.5)

15

There remains the second term, which contributes a nonzero integer:

l∑i=1

l∏j=1j 6=i

(ζpi − ζ

pj)(

1− ζp−i−j)

−1

·l−1∑g=1

(N(g, p, n)−N(l, p, n))(ζpig + ζ

pi(−g)

)

=l∑

i=1ζp−i(l−1) (1− ζ

p−2i)(1− ζ

p−i)·1

p·l−1∑g=1

(N(g, p, n)−N(l, p, n))(ζpig + ζ

pi(−g)

)

=1p·l∑

i=1

l−1∑g=1

(N(g, p, n)−N(l, p, n))[ζp−i(l−g−1) + ζ

p−i(l+g−1) + ζ

p−i(l−g+2)

+ζp−i(l+g+2) − ζ

p−i(l−g) − ζ

p−i(l+g) − ζ

p−i(l−g+1) − ζ

p−i(l+g+1)

]

=1p·l−1∑g=1

(N(g, p, n)−N(l, p, n))l∑

i=1

[ζp−i(l−g−1) + ζ

pi(l−g+2) + ζ

p−i(l−g+2)

+ζpi(l−g−1) − ζ

p−i(l−g) − ζ

pi(l−g+1) − ζ

p−i(l−g+1) − ζ

pi(l−g)

]

=1p·l−1∑g=1

(N(g, p, n)−N(l, p, n))p−1∑i=1

[ζpi(l−g+2) + ζ

pi(l−g−1) − ζ

pi(l−g+1) − ζ

pi(l−g)

]

=1p·l−1∑g=1

(N(g, p, n)−N(l, p, n)) · ε , (2.6)

where ε = 0 if g 6= l − 1 and ε = p if g = l − 1.

Thus, the right-hand side of 2.4 is an integer, and so 2.4 is a polynomial of degree

p− 1 in ζp

over the integers. We can particularly evaluate

p−1∑b=0

NFl(b, p, n)ζ

pb = N(l − 1, p, n)−N(l, p, n) . (2.7)

From the properties of primitive roots, the theorems and corollaries follow.

16

The behavior of ε explains theorem 2. Work of Atkin and Swinnerton-Dyer [4]

yields the arithmetic progressions mentioned by Andrews, for p = 5 and p = 7, in which

the difference N(l − 1, p, pn + d) − N(l, p, pn + d) is identically 0 and equidistribution

of the l-ranks is achieved. A study of the difference N(l − 1, p, n) −N(l, p, n) has been

made for additional prime p by Atkin and collaborators Hussain [3] and O’Brien [8]:

specifically p = 11, 13, 17, and 19.

2.2 Nonprime Moduli

We now turn to a deeper examination of nonprime p. No longer is the polynomial

1 + x + x2 + · · · + x

p−1 irreducible over the integers, so the populations of the various

divisor-groups of residue classes mod p are no longer necessarily equal. However, if we

can establish N(0, p, n) −N(d, p, n) for all d | p, we can state a congruence theorem for

Dl(n) modulo p.

We do this by observing the behavior of Rl(ζpd, ζp2d, . . . , ζ

pld). From Theorem

9 of [1], we have

Rl(ζpd, ζp2d, . . . , ζ

pld) =

∑n≥0

qn ×

∑j | pp | dj

NFl(j, p, n)

p

j

∏k prime

k | pj

(1− 1k)

+∑j | pp - dj

NFl(j, p, n)µ (gcd(p, dj))

gcd(p, dj)j

∏k prime,k|p/jk - p/gcd(p,dj)

(1− 1k)

(2.8)

17

where µ is the standard Mobius function. (The expression appears involved, but calcula-

tion for any given p is not difficult. By way of example we use later, R4(ζ9, ζ92, ζ9

3, ζ9

4; q) =∑n≥0 q

n(NF4(0, 9, n) + 0 ·NF4(1, 9, n)−NF4(3, 9, n)

), andR4(ζ9

3, ζ9

6, ζ9

9, ζ9

12; q) =∑n≥0 q

n(NF4(0, 9, n)− 3NF4(1, 9, n) + 2NF4(3, 9, n)

).)

Calculating this value for each d strictly dividing p gives us a system of d(p)− 1

linear equations in the N(d, p, n) (where d(p) is the divisor function) that we can solve

explicitly for the differences N(0, p, n)−N(d, p, n).

The primary obstacle to this calculation is that we cannot simply assign xi= ζ

pdi

in Theorem 7 of [1], as we did with d = 1 in the theorem above. Doing so produces sin-

gularities in the terms 1(xi−xj)(1−x

−1i x−1

j )when j ≡ ±i (mod p/d). These singularities

are, of course, removable; the problem of evaluation is simply to do so, and the method

is repeated application of L’Hopital’s rule.

The case p = 9 is the first opportunity to employ the method, the most tractable

to calculate explicitly for illustrative purposes, and an interesting example in its own

right.

We begin with Theorem 7 itself:

R4(x1, x2, x3, x4; q) =k∑i=1

R1(xi; q)

k∏j=1j 6=i

(xi− x

j)(1− x−1

ix−1j

)

.

We know that

18

R4(ζ9, ζ92, ζ9

3, ζ9

4; q) =∑n≥0

qn(NF4(0, 9, n)−NF4(3, 9, n)

)

=∑n≥0

qn (N(3, 9, n)−N(4, 9, n)) . (2.9)

Already we can state an interesting congruence: a conjecture of Richard Lewis

[7] proved by Nicholas Santa Gadea [11] states that N(3, 9, 3n) = N(4, 9, 3n). Thus

NF4(0, 9, 3n) = NF4(3, 9, 3n) = NF4(6, 9, 3n) and, since NF4(i, 9, n) = NF4(j, 9, n) for

the 6 residue classes 3 - i, j, we have

Theorem 3. D4(3n) ≡ 0 (mod 3).

To say more regarding the behavior of D4 (mod 9), we need to know the difference

NF4(0, 9, n)−NF4(1, 9, n). To obtain this we wish to calculate, for d = 3,

R4(ζ93, ζ9

6, ζ9

9, ζ9

3; q) = R4(ζ3, ζ32, 1, ζ3; q)

=∑n≥0

qn(NF4(0, 9, n)− 3NF4(1, 9, n) + 2NF4(3, 9, n)

)(2.10)

in terms of R1(ζ3; q).

Our strategy is to replace, one by one, each of the xi

by functions of x1 which

replicate the relations of the ζ3i: x4 by x1, x3 by 1, and x2 by x

−1

1. At each step we

obtain a small number of singularities we can remove. First, let us replace x4 by x1.

19

R4(x1, x2, x3, x1; q) =lim

x4 → x1R4(x1, x2, x3, x4; q)

=lim

x4 → x1

R1(x1; q)

(x1 − x2)(x1 − x3)(x1 − x4)(1− x−11x−12

)(1− x−11x−13

)(1− x−11x−14

)

+R1(x4; q)

(x4 − x1)(x4 − x2)(x4 − x3)(1− x−14x−11

)(1− x−14x−12

)(1− x−14x−13

)

+

R1(x2; q)

(x2 − x1)2(x2 − x3)(1− x−12x−11

)2(1− x−12x−13

)

+R1(x3; q)

(x3 − x1)2(x3 − x2)(1− x−13x−11

)2(1− x−13x−12

)

=R1(x2; q)

(x2 − x1)2(x2 − x3)(1− x−12x−11

)2(1− x−12x−13

)

+R1(x3; q)

(x3 − x1)2(x3 − x2)(1− x−13x−11

)2(1− x−13x−12

)

+lim

x4 → x1

1

(x4 − x1)(1− x−14x−11

)

1∏i=1,4j=2,3

(xi− x

j)(1− x−1

ix−1j

)

×(R1(x4; q)(x1 − x2)(x1 − x3)(1− x−1

1x−1

2)(1− x−1

1x−1

3)−

R1(x1; q)(x4 − x2)(x4 − x3)(1− x−1

4x−1

2)(1− x−1

4x−1

3)))

. (2.11)

After differentiation and taking the limit, we obtain

20

R4(x1, x2, x3, x1; q) =R1(x2; q)

(x2 − x1)2(x2 − x3)(1− x−12x−11

)2(1− x−12x−13

)

+R1(x3; q)

(x3 − x1)2(x3 − x2)(1− x−13x−11

)2(1− x−13x−12

)

+∂∂x1

R1(x1; q)

(x1 − x2)(x1 − x3)(1− x−11x−12

)(1− x−11x−13

)(1− x−21

)

−R1(x1; q)( 1

x1−x2+ 1x1−x3

+x−21x−12

1−x−11 x−1

2

+x−21x−13

1−x−11 x−1

3

)

(x1 − x2)(x1 − x3)(1− x−11x−12

)(1− x−11x−13

)(1− x−21

). (2.12)

For the next step we replace x3 by 1. In the case of d = 3, replacing xp/3 by 1

produces no singularities, and so we need not differentiate. (This is the only divisor where

this degeneracy ever occurs; for any other potential divisor of p, b ldc > 2 means that this

replacement step would produce singularities in the denominator factors (xkpd

− xhpd

)

and (1− x−1kpd

x−1hpd

).) For p = 9, we obtain

21

R4(x1, x2, 1, x1; q) =R1(x2; q)

(x2 − x1)2(x2 − 1)(1− x−12x−11

)2(1− x−12

)

+R1(1; q)

(1− x1)2(1− x2)(1− x−11

)2(1− x−12

)

+∂∂x1

R1(x1; q)

(x1 − x2)(x1 − 1)(1− x−11x−12

)(1− x−11

)(1− x−21

)

−R1(x1; q)( 1

x1−x2+ 1x1−1 +

x−21x−12

1−x−11 x−1

2

+x−21

1−x−11

)

(x1 − x2)(x1 − 1)(1− x−11x−12

)(1− x−11

)(1− x−21

). (2.13)

It remains to replace x2 by x−1

1.

R4(x1, x−1

1, 1, x1; q) =

lim

x2 → x−11

R(x1, x2, 1, x1; q) =R1(1; q)

(1− x1)3(1− x−11

)3

+lim

x2 → x−11

1

(x1 − x2)2(x1 − 1)(1− x−11x−12

)2(1− x−11

)(1− x−12

)(1− x−21

)(x2 − 1)

×R1(x2; q)((x1 − 1)(1− x−1

1)(1− x−2

1)) + ((x1 − x2)(1− x−1

1x−1

2)(x2 − 1)(1− x−1

2))

×

∂

∂x1R1(x1; q)−R1(x1; q)(

1x1 − x2

+1

x1 − 1+

x−2

1x−1

21− x−1

1x−12

+x−2

11− x−1

1

)

.

(2.14)

22

We differentiate (twice) with respect to x2 and note the identity

lim

x2 → x−11

∂2

∂x22R1(x2; q) = x1

4 ∂2

∂x12R1(x1; q) + 2x1

3 ∂

∂x1R1(x1; q)

to obtain in the limit

R4(x1, x−1

1, 1, x1; q) =

R1(1; q)

(1− x1)3(1− x−11

)3+

−x1−4

2(1− x1)3(1− x1−1)3(1− x1

−2)3

×

∂2

∂x12R1(x1; q)x1

4(x1 − 1)2(1− x1−1)

2(1− x1

−2)

+ 2∂

∂x1R1(x1; q)(1− x1)2(1− x−1

1)2(−x1

3 − 2x12 − 2x1)

+ 2R1(x1; q)(1− x12)

2(1− x1

−2)]

. (2.15)

We have now removed all the troublesome singularities and can set in the last

identity x1 = ζ3 to evaluate

23

R4(ζ3, ζ32, 1, ζ3; q) =

∑n≥0

qn(NF4(0, 9, n)− 3NF4(1, 9, n) + 2NF4(3, 9, n))

=R1(1; q)

27+

−ζ32

54(1− ζ3)3

×

[9(ζ3 − ζ3

2)∂2

∂z2R1(z; q)

∣∣∣z=ζ3

+ 18∂

∂zR1(z; q)

∣∣∣z=ζ3

+ 6(1− ζ32)R1(ζ3; q)

]

=154

(2R1(1; q) + 3ζ3

2 ∂2

∂z2R1(z; q)

∣∣∣z=ζ3

+ 2(ζ3 − 1)∂

∂zR1(z; q)

∣∣∣z=ζ3

− 2R1(ζ3; q)

).

(2.16)

We wish to rewrite this formula in terms of the rank classes N(j, n). The termwise

first and second derivatives of N(j, n)zjqn, jN(j, n)zj−1qn and j(j − 1)N(j, n)zj−2

qn

respectively, group themselves thus by the residue class of j modulo 3 when evaluated

at z = ζ3:

24

R4(ζ3, ζ32, 1, ζ3; q)

=154

∑n≥0

qn

n∑k=−n

((27k2 − 3k)N(3k, n)− 6kN(3k + 1, n) + 2N(3k + 2, n)

+ ζ3

((27k2 + 15k)N(3k + 1, n)− (6k + 4)N(3k + 2, n)

)+ ζ3

2 ((27k2 + 33k + 8)N(3k + 2, n)− 6kN(3k, n)))

=154

∑n≥0

qn

n∑k=−n

(27k2 + 3k)N(3k, n)− 6kN(3k+ 1, n)− (27k2 + 33k+ 6)N(3k+ 2, n)

+ ζ3

(6kN(3k, n) + (27k2 + 15k)N(3k + 1, n)− (27k2 + 39k + 12)N(3k + 2, n)

).

(2.17)

Here we pause to observe that we can simplify the sum above by recalling that,

due to the evenness of the rank function, for any j

j∑k=−j

kN(3k, n) =j∑

k=−jkN(3k + 1, n) + (k + 1)N(3k + 2, n) = 0

and

j∑k=−j

k2N(3k + 1, n) =

j∑k=−j

(k + 1)2N(3k + 2, n) .

With these two identities the ζ3 term of 2.17 wholly vanishes. (We knew it must,

since of course R4(ζ3, ζ32, 1, ζ3; q) has integral coefficients.)

25

Upon discarding the vanishing ζ3 term and simplifying the remainder with the

relations above we have that

NF4(0, 9, n)− 3NF4(1, 9, n) + 2NF4(3, 9, n)

=19

n∑k=−n

((9k2

2

)N(3k, n)−

((9k2 + 9k)

2

)N(3k + 2, n)

)

=n∑

k=−n

(k2

2N(3k, n)− k(k + 1)

2N(3k + 2, n)

)

=n∑k=1

k2N(3k, n)− k(k + 1)

2(N(3k + 1, n) +N(3k + 2, n)) . (2.18)

Thus, since we already know NF4(0, 9, n)−NF4(3, 9, n) = N(3, 9, n)−N(4, 9, n),

NF4(0, 9, n)−NF4(1, 9, n) = −23

(N(3, 9, n)−N(4, 9, n))

+13

n∑k=1

k2N(3k, n)− k(k + 1)

2(N(3k + 1, n) +N(3k + 2, n)) . (2.19)

Putting these all together, we have

26

D4(n) = NF4(0, 9, n) + 6(NF4(0, 9, n)− (NF4(0, 9, n)−NF4(1, 9, n)))

+ 2(NF4(0, 9, n)− (NF4(0, 9, n)−NF4(3, 9, n)))

= 9NF4(0, 9, n) + 2(N(3, 9, n)−N(4, 9, n))

−n∑k=1

2k2N(3k, n)− k(k + 1)(N(3k + 1, n) +N(3k + 2, n))

≡ 2(N(3, 9, n)−N(4, 9, n))

−n∑k=1

2k2N(3k, n)− k(k + 1)(N(3k + 1, n) +N(3k + 2, n)) (mod 9) . (2.20)

For n ≡ 0, 1, 2 (mod 3), the identities of [11] provide specializations of this identity

when we dissect the sum over k by the residue classes of k modulo 3.

In the case of general p and divisor d, we perform variable replacements patterned

on those we saw above. We replace xi+kp

d

with xifor 0 < i < p

d , replace xkpd

with 1, and

finally replace xpd−i

with x−1

ifor 0 < i ≤ b p2dc. We eventually encounter derivatives

of order up to 2d, in order to clear singularities. When we then evaluate Theorem 7 at

xi

= ζpdi, a great deal of simplification can occur by working with the evenness of the

rank function. A general form for these functions should be easy to obtain.

27

Chapter 3

Generalizing Sylvester’s Bijection

One of the first theorems the student of partitions learns is Euler’s theorem that

the number of partitions of n into odd parts equals the number of partitions of n into

distinct parts. This simple statement has been refined, generalized, and expanded upon

by many workers in the field. In 1883, Glaisher showed this to be the m = 2 case of

a general theorem equating the number of partitions of n into parts not divisible by m

to partitions into parts appearing fewer than m times. J.J. Sylvester’s student Franklin

made it the k = 0 case of a theorem equating the number of partitions of n with k sizes

of even part, and those with k sizes of repeated part.

In 1882, Sylvester published his famous paper [14], containing his bijective proof

of Euler’s theorem. Later authors – starting with Cayley and more recently including

Bessenrodt [5], Fine [6], Kim and Yee [15], and Pak and Postnikov [10], to name a few

– have found refinements of Euler’s theorem via this bijection. The transformation not

only maps partitions with odd parts to those with distinct parts, but also preserves a

number of statistics on partitions of each type, such as the number of parts in the starting

odd partition λ and the ”alternating length” of the target 2-distinct partition µ (defined

as µ1−µ2+µ3−µ4+ . . . , and easily seen to be number of odd parts in the conjugate µ).

The introduction of these statistics allows the construction of finer identities equating

the generating functions of partitions with parameters for each statistic.

28

The challenge this chapter sets is to combine these two lines of development, and

carry the refined statistics present in Sylvester’s bijection from partitions into odd parts

toward partitions into parts not divisible by m.

We begin with an identity due to N.J. Fine. In [6], he proves that:

Theorem 4. (Fine) Partitions of n into distinct parts with largest part M are equinu-

merous with partitions of n into odd parts with largest part plus twice the number of parts

2M + 1.

Theorem 4 is numbered 23.91 in [6]. The following corollary actually appears

earlier, as equations 23.8, but Fine points out that it can be deduced from 23.91 by

noting that the number of parts in a partition of n into odd parts has the same parity

as n:

Corollary 3. The number of partitions of n into distinct parts with largest part

≡ a (mod 2), a = 0, 1, is equal to the number of partitions of n into odd parts with

largest part ≡ 2a+ 1 (mod 4) if n is even, and ≡ 2a− 1 (mod 4) if n is odd.

Fine proved his theorem analytically, but Pak and Postnikov in [10] show that

this, and additional statistics, can be proved with a generalization of Sylvester’s bijection.

(They also generalize these statistics to the (m, c) case, as we will discuss momentarily.)

The most general collection of statistics appears to be in a paper of Zeng [16] published

in 2005. Denote by l(λ) be the number of parts in λ a partition into odd parts and by

d2(λ) the largest i such that (λi− 1)/2 ≥ i. Then from that paper, we have:

Theorem 5. Let λ be a partition of n into odd parts and φ be Sylvester’s bijection,

µ = φ(λ). Then µ is a partition of n into distinct parts; l(λ) equals the alternating

29

length of µ; l(λ) + (λ1 − 1)/2 = µ1; the number of distinct parts in λ equals the number

of sequences of consecutive parts in µ; and d2(λ) = bl(µ)/2c.

The one remotely successful attempt at generalization of Sylvester’s bijection to-

ward Glaisher’s theorem appears to be the (m, c)-analogues of the bijection. These treat

what are called (m, c) partitions – those in which parts are all congruent to c (modm).

These are equinumerous with partitions of type (c,m−c, c,m−c, ...), where a partition is

of type (a1, a2, . . . ) if the largest part appears a1 times, the next largest part appears a2

times, etc. There are also sets of related statistics: the same as in the previous theorem,

except that in general we replace 2 with m, such as using dm

(λ) (replacing (λi− 1)/2

with (λi− c)/m) and alternating length µ1 − µm + µ

m+1 − µ2m + . . . . (Readers of

Zeng’s paper should be careful about the latter, which is not clarified.) However, this

certainly falls short of treating all partitions into parts 6≡ 0 (modm).

More importantly, Zeng points out that the identity thus obtained (Theorem 4

in [16]) is algebraically equivalent to the original identity with a simple substitution of

variables. The reason for this is that, with regard to the characteristics manipulated

by the bijection, any (m, c) partition has the same m-modular shape regardless of what

the m and c actually are. An algebraically richer identity thus requires considering

partitions into parts not divisible by m in which different residues of parts modulo m

appear. In this chapter are established such identities in the cases where residues mod

m increase weakly monotonically from the smallest part to the largest, and those in

which the residues decrease monotonically. To be specific, we establish, where fλ

is the

number of descents among the nonzero residues (modm) of parts of λ read right to left

as a multiset on [1, . . . ,m− 1]:

30

Theorem 6. The number of partitions µ of n into parts appearing fewer than m times,

with largest part µ1, is equal to the number of partitions λ of n into parts not divisible

by m with λ1 +m ∗ (l(λ)− fλ) = m ∗ µ1 + j, 0 < j < m.

Summation in residue classes gives us

Corollary 4. The number of partitions µ of n into parts appearing fewer than m times,

with largest part µ1 ≡ b (modm), is equal to the number of partitions λ of n into parts

not divisible by m with λ1 +m ∗ (l(λ)− fλ) ≡ mb+ j (modm2), 0 < j < m.

We actually get more (for the full statement, see Theorem 9), though the definition

and counting of chains of consecutive-or-equal parts requires some delicacy. The (m, c)

generalization of Sylvester’s bijection can stand up to most of the task, but runs into

difficulties when calculating the number of these sequences. We therefore use a different

generalization; the next section lays out the definition and the necessary tools for its

analysis.

3.1 Definitions

In his 1981 thesis [13], Dieter Stockhofe constructed for each n a collection of

bijections Lm,n

which together generate all bijections on the set of partitions of n. Each

Lm,n

is itself a collection of bijections between certain classes of partitions of n with

specific characteristics related to their m-modular diagrams. To describe the Lm,n

more

precisely we need the following definitions, mostly from [13]. Fixing α a partition of n,

α = (α1, . . . , αk), and a modulus m:

Definition 4. The m-weight of αi

is |αi|m

= bαim c.

31

We will illustrate the definitions of this section with a continuing example partition

α = (42, 39, 30, 25, 23, 20, 16, 10, 7, 5, 5), using the modulus m = 5 unless otherwise noted.

Example 1. |α1|5 = 8; |α11|5 = 1.

Definition 5. Let βi≡ α

i(modm) be the least nonnegative residue of α

i. Then

the residue-vector ρ(α) is the r-tuple (ρ1, . . . , ρr) = (βi1, . . . , β

ir) of nonzero β

iwith

i1 < i2 < · · · < ir.

Intuitively, the process to construct ρ consists of removing all parts of α divisible

by m and reducing the remaining parts mod m.

Example 2. ρ(α) = (2, 4, 3, 1, 2).

Definition 6. A part αiof α contains j m-edge units (or simply edge units when m is

understood) if (m+ 1)j > αi− α

i+1 ≥ mj, setting αk+1 = 0.

Intuitively, j measures the amount by which αi

exceeds the minimum multiple of

m necessary for α to be a partition.

Example 3. α has 5-edge units in parts 2, 3, 7, and 11. Observe that the last part

contains a 5-edge unit since it is at least 5. Also note that it is possible for a part to have

more than one m-edge unit; for example, if we had set m = 2, then α2 would contain 4

2-edge units, since α2 − α3 = 9.

In an associated vein, we may speak of an m-strip: a set consisting of the m-edge

unit in αi

and one multiple of m in each larger part. These m-strips can be subtracted

from the parts in α in which they appear, leaving a sequence which is still a partition

(m-edge units are defined by being in excess of the minimum allowable size of a part)

32

and replaced in the new partition as, for example, a part equal to m times the number

of elements in the strip so defined. Reversible examples of such manipulations are used

by Stockhofe in defining Lm

.

Example 4. The 5-strips of α are of lengths 2, 3, 7, and 11; there is one of each length.

If the set of all m-strips is subtracted from α and collected as a new partition

into parts divisible by m, this new partition is called mαs, the strips of α. (Addition

and scalar multiplication of partitions are defined componentwise as the standard vector

operations for vectors of length equal to the longer partition, filling out the smaller with

zeroes; subtraction is defined when the result is still a partition. Once m is fixed we

often speak of αs

and mαs

interchangeably as suits the context.)

Example 5. αs

= (20, 20, 15, 10, 10, 10, 10, 5, 5, 5, 5). It is perhaps more illuminating to

observe that αs′ = (55, 35, 15, 10) = 5(11, 7, 3, 2).

Definition 7. Pρk,l

is the set of partitions that have residue-vector ρ, k parts divisible by

m, and a total of l m-edge units (or, equivalently, m-strips). Pρk,l

(n) is the set of such

partitions of n.

With these definitions, Lm,n

exchanges Pρk,l

(n) with Pρ

l,k(n). It thereby estab-

lishes a generalization of conjugation (which is, in fact, L1,n) for m-modular diagrams.

Later we will consider the actual bijection; for many counting theorems, it is sufficient

to recall the somewhat astounding fact that these two classes are equinumerous.

In order to use this generalization of conjugation to extend classical identities, we

now make a simple additional observation: namely, that when αi

= βi+ t

im contains

j m-edge units, if βi≥ β

i+1 then j = ti− t

i+1, whereas if βi< β

i+1, j = ti− t

i+1 −

33

1. Viewing ρ as a permutation of a multiset in [1, . . . ,m− 1], the number of times

ρi< ρ

i+1 – the number of descents in the word ρ, to use a more common phrasing in

combinatorics – is a statistic that relates further useful information about the partition.

It is also interconnected with other statistics: for example, each descent increases by 1

the minimum possible m-weight of the largest part of α, and decreases (in a suitably

defined average sense) by m−1m−2 the number of parts in ρ.

The observer will notice that m−1m−2 is undefined for m = 2 and, indeed, makes

little sense for m = 1. Naturally, in these cases there are no descents in ρ: for m = 1,

ρ is empty, and for m = 2, ρ = 1r. It is this simplicity that hides the more elaborate

combinatorial structure of the generalizations we deal with here.

Finally, we introduce some terminology to smooth the description of the theorems

we produce.

Definition 8. Call⋃ρ,lPρ

0,l(the set of partitions into parts not divisible by m) m-odd

partitions, and⋃ρ,k

Pρ

k,0(the set of partitions in which there are no m-strips, or, that

have first differences less than m and smallest part less than m) m-flat partitions.

Applying traditional conjugation to m-flat partitions we see that these are equinu-

merous with the set of partitions in which parts appear fewer than m times, and as this

gives Glaisher’s generalization of Euler’s odd-distinct theorem it seems useful to denote

these as m-distinct partitions.

(The reader should be alerted that the terminology ”flat” is used in at least

one other place the author knows of, in Sloane’s encyclopedia of integer sequences [12];

34

there, ”flat partitions” (Sequence A034296) are partitions with first differences less than

or equal to 1 without restriction on the size of the smallest part.)

The remaining portion of α, called the flat part of α, can be broken down as

mρOmα

fwhere mρ is the unique partition in P

ρ

0,0and α

fis defined by the following

procedure: first, remove from the flat part of α any repeated parts of size divisible by m,

but not the last such part (say, part i) of any given size if it lies between parts such that

the residues βi−1 ≥ βi+1. Each removed part becomes a part of α

f. For the remaining

parts divisible by m, remove the intervening parts and prevent m-strips from appearing

by removing ”angles”: working from the largest part divisible by m, remove the part

itself and conjoin it with the m-strip that results from its removal to form a single part

of αf. The operation O is the (unique) reverse of this process.

Example 6. Let m = 5, α = (42, 39, 30, 25, 23, 20, 16, 10, 7, 5, 5). Then ρ = (2, 4, 3, 1, 2),

mρ = (12, 9, 8, 6, 2), α

s= (20, 20, 15, 10, 10, 10, 10, 5, 5, 5, 5), and α

f= (25, 25, 15, 5).

The operation O inserts the parts of αf

into mρ in this order: first to insert a weight of

25, 5 is added to parts 12, 9, and 8 of mρ to make parts 17, 14, and 13; then a new part

10 is added between 13 and 6. Second, to insert the next 25, 5 is added to parts 17 and

14 to make parts 22 and 19; then a new part 15 is added between 19 and 13. Finally, 15

and 5 are added as parts on their own to create mρOαf

= (22, 19, 15, 15, 13, 10, 6, 5, 2),

the flat part of α.

35

Fig. 3.1. α decomposed as in Example 6.

36

When one of αf

or αs

is empty – that is, when α is an m-odd or m-flat partition

respectively – then the bijection Lm,n

exchanges Pρk,l

(n) with Pρl,k

(n) thus:

Lm,n

((mρOmαf) +mα

s) = (mρOmα

s′) +mα

f′.

Our considerations are solely on relations between m-odd and m-distinct parti-

tions, so this description of the simplified behavior of Lm,n

on such partitions suffices for

our purposes. However, the map is well-defined for all partitions of n, and the interested

reader is directed to [13], or the translation in Appendix B, for a fuller description and

complete detail of exposition.

3.2 Sylvester’s Map

In [9], Pak notes that in the literature discussing partitions into odd and distinct

parts, numerous bijections appear... and are almost universally the same as that given by

Sylvester in his original bijective proof of Euler’s theorem. When we observe, therefore,

that the composition of Stockhofe’s bijections L1,nL2,n is precisely Sylvester’s bijection,

it may seem to be gilding the lily. However, when we note that L1,n Lm,n serves the

same purpose for the (m, c)-analogues (the same map suffices for all c for each m),

something new finally happens: for while L1,n Lm,n is precisely the so-called fishhook

bijection for partitions wherein the residues of parts mod m are all the same, it is not

the same map for partitions in which residues differ.

To prove these claims we will employ the version of Sylvester’s bijection labeled η

in [9]. The definition I construct below for general residue-vector ρ simplifies to become

the definition so named that Pak uses when applied to odd partitions, and the graphical

fishhook map that Zeng uses when applied to (m, c) partitions.

37

Definition 9. Fix a modulus m. Let λ be a partition of n with parts 6≡ 0 (modm),

ρ = (ρ1, . . . , ρk). Then η(λ) = (. . . , ηmt+i, . . . ), where with 0 ≤ i < m, 0 < t ≤ d

m(λ),

ηmt−i := |λ

t|m− t+ #λ

r| |λ

r|m≥ t − (t− 1)

+ #ρa|(a = t or |λ

a|m

= t− 1) and (m− i ≤ ρa),

and for t = dm

(λ) + 1, ηmt+i := #ρ

a| a ≥ t and |λ

a|m

= t− 1 and (m− i ≤ ρa).

Fig. 3.2. The generalized η: η((17, 14, 13, 12, 6)) = (8, 8, 7, 7, 7, 5, 4, 4, 4, 3, 2, 2, 1)

Recall that for a partition into parts not divisible by m, dm

(λ) is the largest c

such that bλcm c ≥ c. Another way to interpret this is by removing the residue-vector ρ

and regarding the m-modular diagram below it, with the m replaced by the dots of a

38

typical Ferrers diagram: dm

(λ) is then the size of the Durfee square of that partition.

On the other hand, dm∗(λ) uses the largest c such that dλc

m e ≥ c – it simply includes

the residue-vector.

In the example above, we have dm

(λ) = 2 and dm∗(λ) = 3. The two values differ

exactly when |λdm(λ)+1|m

= dm

(λ), when there is an entry of the residue-vector at the

corner of the m-modular Durfee square.

To read η(λ) off of its m-modular diagram [λ]m∗, we draw hooks through the

main diagonal and read their lengths. Here we have

[(17, 14, 13, 12, 6)]5∗ =

5 5 5 5 5

5 5 5 5 1

5 4 3 2

2

.

Now observe the definition of η. Parts appear in groups of m, in this case 5. In

the first hook, we begin with t = 1, and construct parts 1 through 5. The 5-weight of λ1

is 3, from which we subtract 1 so as not to overcount the corner of the hook; there are 5

parts of 5-weight at least 1, from which we subtract t−1 parts so as not to recount parts

larger than λ1 (of course, there are none yet). So far, we have 3-1+5-1=7. There is one

ρa

with a = t, i.e., ρ1 = 2. There are no parts with |λa|m

= 0. So there are 2 parts in

η(λ) of size 1 greater than 7: thus, the first 5 parts are (8, 8, 7, 7, 7). These correspond

to the 5 outermost hooks in the diagram above.

One can immediately see from this description why the largest part of η(λ) will

have size equal to the m-weight of the largest part of λ, plus the number of parts of λ –

39

or, if we scale by m, that m times the largest part of η will be roughly λ1 plus m times

the number of parts of λ.

The second group of 5 parts is constructed similarly. The 5-weight of λ2 is 2; we

subtract 2, one to not overcount the corner and one so as not to recount the first row;

there are 4 parts of 5-weight at least 2, from which we ignore the first part, so these five

parts are of size at least 2-2+4-1=3. There is one residue for λ2, which is 4, and one

residue of a part with 5-weight exactly 1, namely ρ5 = 1. Thus, the first 4 parts are of

size at least 1 greater than 3, and the first of the five is 2 greater: the first 10 parts of

η(λ) are (8, 8, 7, 7, 7, 5, 4, 4, 4, 3).

Finally, the last group is different since we are at the outer corner of the 5-

modular Durfee square. The last group of 5 (or fewer) parts counts the number of

remaining residues on parts with 5-weight 2 of size at least i: our residues are 3 and

2, so we have 3 parts of size at least 1, and another 2 of size at least 2. Thus, η(λ) =

(8, 8, 7, 7, 7, 5, 4, 4, 4, 3, 2, 2, 1).

We can now prove that L1,n Lm,n(λ) = η(λ) for (m, c) partitions. The proof

illuminates several aspects of the behavior of L1,n Lm,n that will be useful to recall in

the more general case.

Theorem 7. Given a modulus m and a partition λ with ρ = (c, c, . . . , c),

L1,n Lm,n(λ) = η(λ).

Proof. If dm

(λ) ≥ 1, the largest c parts of η(λ) are η1 = · · · = ηc

=

|λ1|m − 1 + #λr| |λ

r|m≥ 1 − 0 + #ρ

a|(a = 1 or |λ

a|m

= 0) = |λ1|m + k , and if

dm

(λ) = 0 then η1 = · · · = ηc

= #ρa| a ≥ 1 and |λ

a|m

= 0) = |λ1|m + k (= k) .

40

If dm

(λ) = 0, then Lm,n

(λ) = λ and so φ = L1,n Lm,n(λ) = (k, . . . , k) = (kc).

If dm

(λ) ≥ 1, then φ1 = · · · = φc

= l(Lm,n

(λ)′) = l(λs) + l(ρ) = |λ1|m + k. So the first

c parts match. If dm

(λ) = 0, then there are no more parts and we are done.

If dm

(λ) ≥ 1, the next m − c parts of η(λ) are ηc+1 = · · · = η

m= |λ1|m − 1 +

#λr| |λ

r|m≥ 1 − 0 + 0 = l(λ

s)− 1 + λ

s1. The next m− c parts of φ are the index of

the second-largest part in Lm,n

(λ), which resulted from inserting the largest part of λs

at the beginning of the O operation. This insertion produced a part of size m at index

λs1

, adding m to each previous part; the remaining l(λs)− 1 parts of λ

swere inserted

at various lower indices. Thus the index of the last m in Lm,n

(λ), which becomes the

value of the parts φc+1 = · · · = φ

m, is l(λ

s)− 1 + λ

s1.

If dm

(λ) = 1, the next c parts of η(λ) have t = dm

(λ) + 1 and so are ηm+1 =

· · · = ηm+c = #ρ

a| a ≥ 2 and|λ

a|m

= 1 = λs1− 1. In this case as well, the next c

parts of φ are the index of the largest of the parts m+ c which resulted from O adding

m to the first λs1− 1 entries, before inserting any number of parts m to the right of

those entries. (It is possible, of course, that λs1

= 1, i.e. λ = l(λs)m+ c, in which case

all of these entries are 0.) Thus η2m−i = φ2m−i and, as these are the last entries, we

are done.

The next case should be sufficient to illuminate the pattern. If dm

(λ) ≥ 2,

ηm+1 = · · · = η

m+c = |λ2|m − 2 + #λr| |λ

r|m≥ 2 − 1 + #ρ

a|(a = 2 or |λ

a|m

= 1)

= |λ2|m − 2 + #λr| |λ

r|m≥ 1. On the other hand, φ

m+1 = · · · = φm+c are the

index of the third-largest part of Lm,n

(λ), which is the part m + c resulting from the

addition of m to each part c of mρ of index less than λs1

. Such an m was added to the

first λs1− 1 parts. The operation O then inserted parts of sizes divisible by m, once

41

for each remaining part of λs: λ

s2, λ

s3, etc. Those strips of sizes ≥ 2 were inserted

as parts of size ≥ 2m; those of size 1 were inserted as parts of size m, and did not

increase the index of parts m + c. The number of those strips of size ≥ 2, barring the

previously-inserted λs1

, is precisely |λ2|m − 1. Thus, the indices of parts of size m + c

are exactly λs1− 1 + |λ2|m− 1 = |λ2|m− 2 + #λ

r| |λ

r|m≥ 1, and so φ

m+1 = ηm+1.

The analysis continues along these lines until the Durfee square of λs

is exhausted.

At each point the sizes of parts are equal.

Example 7. However, note that for m = 3, λ = (8, 8, 4, 1) has ρ = mρ = (2, 2, 1, 1),

λf

= () and λs′ = (9, 6). We then see that η(λ) = (6, 5, 4, 3, 2, 1), while L1,21L3,21(λ) =

L1,21((6, 5, 5, 3, 1, 1)) = (6, 4, 4, 3, 3, 1).

Previous authors do not seem to have had success generalizing Sylvester toward

Glaisher with the bijection η. Most of the required statistics can be established; it is the

number of sequences of consecutive parts that appears to pose the challenge. In light of

the example above, then, we might hope that analysis of the characteristics of partitions

and their images under L1,n Lm,n will provide a better route for generalization.

Let us begin by noting the two features of general partitions into parts not divisible

by m which are degenerate in the (m, c) case: first, where residues differ, λ possesses

distinct parts that arise without a distinct length of strip in λs, which was the sole source

of distinct parts in the (m, c) case. Second, in the (m, c) case mρ = ρ = (c, c, . . . , c)

since the residue-vector is itself a partition. If however ρ possesses any descents – that

is, ρi< ρ

i+1 for any i – then mρ 6= ρ, and contains parts of nonzero m-weight. For

42

example, if m = 3, ρ = (1, 2), then mρ = (4, 2). This affects the behavior of the statistics

cited in Theorem 5.

But while η is sensitive to the residues ρi, we can observe that when ρ possesses

no descents – that is, ρ1 ≥ ρ2 ≥ · · · ≥ ρk

– then the behavior of O and so Lm,n

is

essentially blind to the actual values of ρ. The insertion process works in the same way

as in the (m, c) case. This is the first kind of partition we discussed earlier. Let us label

them for the sake of conciseness in stating the theorem:

Definition 10. If λ = (λ1, . . . , λk) is a partition of n with no part divisible by m, and

residue-vector ρ = (ρ1, . . . , ρk), ρ1 ≥ · · · ≥ ρk, then call λ an m-falling partition of n.

As one might suspect from the points of difference from the (m, c) case described

above, we must modify our notion of the number of distinct parts:

Definition 11. With λ as above, call nd(λ) the number of distinct kinds of residues in

ρ plus the number of distinct sizes of m-strips in λs, not counting any m-strip of length

k (i.e., if the smallest part of λ is greater than m), nor any strips of size i where there

exists a strip such that(λs′)i− i is exactly the index of the last appearance in ρ of a ρ

j

of a given size.

Example 8. For m = 3, λ = (8, 8, 4, 1) as above, λ is a 3-falling partition. It has 2

distinct kinds of residues, and 2 distinct sizes of m-strips. The longest m-strip is not of

length 4. While(λs′)

1− 1 = 2, the index of the last appearance of the residue 2 in ρ,

there are no strips of size 1 to skip, so nd(λ) = 4.

When λ is an (m, c) partition, nd(λ) is exactly the number of distinct sizes of

parts in λ: there is 1 distinct residue, and any distinct length of strip gives a distinct

43

part of λ, unless the very first strip is of length k itself; nor are there any strips of the

special sizes we decline to count. For general m-odd partitions, nd(λ) is very nearly the

number of distinct sizes of parts, underestimating by the number of missed special parts

and overestimating by the count of indices where new sizes of λ coincide with the last

appearance of any given size of ρj.

Definition 12. With µ a partition into parts appearing fewer than m times, let nm

(µ)

be the number of maximal-length sequences in µ, of type (c,m− c, c,m− c, . . . ) for any

given c, of parts differing by at most 1. Call such sequences m-chains.

The targets of the fishhook bijections from (m, c) partitions have all their se-

quences of such a type, starting with the same c or m− c. Hence this definition special-

izes to that of the previous literature’s nc(µ) for such partitions, and to ”sequences of

consecutive parts” for the odd-distinct case.

Example 9. For m = 3, µ = (6, 4, 4, 3, 3, 1), the first 3-chain is of type (1) and consists

of the part (6). The second 3-chain is of type (2) and consists of the parts (4, 4): while 4

and 3 are consecutive parts, 3 would need to appear exactly once to continue the 3-chain,

making it of type (2, 1, . . . ). Instead, (3, 3) is itself a chain of type (2), and (1) is a chain

of type (1). Thus nm

(µ) = 4.

To shorten our theorems, we define

Definition 13. A partition is of m-alternating type if it is of type (1,m − 1, 1,m −

1, . . . , f(2),m− 2, 2,m− 2, . . . , f(ci),m− c

i, . . . ), in which a given sequence for c

imay

be empty and where f(ci) = c

ior c

i− c

i−1, according as the previous filled entry was

(m−ci−1 or nonexistent) or c

i−1 respectively; it is of m′-alternating type if it is of type

44

(m− 1, 1,m− 1, . . . , f(m− 2), 2,m− 2, 2, . . . , f(m− 3), 3, . . . ) in which a given sequence

for ci

may be empty and where f(m − ci) = m − c

ior m − c

i+ c

i−1, according as the

previous filled entry was (ci−1 or nonexistent) or m− c

i−1 respectively.

Partitions of this type are, of course, a subset of m-distinct partitions.

Finally, a comment on notation, both for convenience and to match the previous

literature. What we have been calling dm

(λ) for m-odd partitions has useful meaning in

terms of the breakdown (mρOmλf) +mλ

s: it is the size of the standard Durfee square

of λs. However, in dealing with (m, c) partitions previous authors found it useful to

include the residues c when considering the Durfee square, which is the domain of the

statistic dm∗(λ). Thus, for an m-odd partition we point out that d

m∗(λ) is d

m(λ) + 1

if |λdm(λ)+1|m

= dm

(λ) – that is, if every m-strip in the Durfee square of λ is of length

strictly greater than the size of dm

(λ – and dm

(λ) otherwise. The statistic dm∗(λ) is

what Zeng, for example, denotes dm

.

Now we can state the following theorem:

Theorem 8. Let λ be an m-falling partition of n. If µ = L1,n Lm,n(λ), then µ is a

partition of n in which:

• µ is of m-alternating type;

• l(λ) = la(µ) = µ1 − µm + µ

m+1 − µ2m + . . . ;

• l(λ) + bλ1m c = µ1;

• dm∗(λ) = b l(µ)+m−1

m c; and

• nd(λ) = n

m(µ) .

45

Furthermore L1,n Lm,n is reversible.

Proof. Let λ be an m-falling partition of n. Then λ ∈ Pρk,0

(n), and so λf

= ∅. In that

case, Lm,n

((mρOm∅) +mλs) = (mρOmλ

s′) + ∅ ∈ Pρ

0,k(n), a subset of partitions of n

in which the smallest part is less than m and the differences between parts are less than

m, and so L1,n Lm,n(λ) =(Lm,n

(λ))′

is a partition of n in which parts appear fewer

than m times.

Parts m, 2m, . . . , dm

(λ)m all appear in Lm,n

(λ): consider in turn the effect of O

on each of the parts of λs

as it inserts the lengths of the m-strips – the parts of λs′ – into

mρ = ρ. The first insertion produces a part of size m at index

(λs′)

1, increasing the

indices of the further parts of mρ by 1, and adds m to each of the previous(λs′)

1− 1

residues to form parts m+ ρa. The next insertion produces a part of size 2m just before

one of the newly produced parts, at index(λs′)

2−1, and adds m to each of the previous(

λs′)

2− 2 parts to form parts of size 2m+ ρ

a, while increasing the index of all further

parts of the diagram in progress, including the previously-inserted part of size m. As

long as(λs′)i− i ≥ 0, a part of size i will be inserted.

We now note that if(λs′)i− i = 0, then i = d

m(λ) and the part is inserted as a

part of size dmi as the largest part of the partition. If on the other hand

(λs′)dm−d

m>

0, the strip is inserted as a part of size dm

(λ) at some index larger than 1, and m added

to all previous parts, so that the largest part of the partition is dm

(λ) + ck

for some

residue 0 < ck< m.

Further parts of λs′ are inserted as repetitions of parts of size mi, . . . ,

m(i − 1), . . . ,m, appearing next to the parts of the appropriate size already inserted

from those m-strips that appeared above the Durfee square of λs.

46

Thus the largest part of Lm,n

(λ), and so l((Lm,n

(λ))′)

= l(µ), is at least

dm

(λ)m and at most dm

(λ)m+ (m− 1). Hence dm

(λ) = b l(µ)m c. Furthermore, because

a part of size m appears in Lm,n

(λ), the number of parts of λ that appear beyond

this part total µ1 − µm

; the number of parts of λ that appear between the part(s)

of size m and the part(s) of size 2m total µm+1 − µ2m; etc., so that l(λ) = l

a(µ) =

µ1 − µm + µm+1 − µ2m + . . . .

The number of parts in Lm,n

(λ), which becomes µ1, is exactly the number of

parts of λ plus one part for every m-strip, i.e. bλ1m c. Thus l(λ) + bλ1

m c = µ1.

Finally, we consider how chains arise in µ. We begin with insertion of parts above

the Durfee square.

If(λs′)i−1

=(λs′)i

with i ≤ dm

(λ), then O inserts(λs′)i−1

and(λs′)i

into

mρ in the process of continuing a sequence in mρ of the form (. . . ,mi+ ρ

j−1,mi,m(i−

1)+ρj,m(i−1),m(i−1)+ρ

j+1, . . . ). If(λs′)i−1

>(λs′)iwith i ≤ d

m(λ), then there

will be a sequence of more than one part in Lm,n

(λ) not divisible by m between mi and

m(i−1): that is, of the form (. . . ,mi,m(i−1)+ρj, . . . ,m(i−1)+ρ

j+a,m(i−1),m(i−1)+

ρj+a+1, . . . ) with a > 0. In the former case, the parts of µ thus created by conjugation

are a sequence of consecutive parts of type . . . , ρj−1,m − ρ

j, ρj,m − ρ

j+1, . . . ). If

ρj

= ρj+1, then the sequence is of type (. . . ,m − ρ

j, ρj,m − ρ

j+1, . . . ); otherwise,

it is a new sequence. Likewise with ρj−1 and ρ

j. In the latter case, wherever at

least two of the ρj, . . . , ρ

j+a are equal, in the conjugate of the sequence contains

nonconsecutive parts so that one new sequence begins, and wherever they are unequal, a

series of consecutive parts appears in µ of typeρj−ρ

j+1, ρj+1−ρj+2, . . . . Since ρi6= m,

each of these new types of part individually constitutes a distinguishable sequence of type

47

(c,m− c, c,m− c, . . . ) of length 1, and the next part, of type m− ρj, also begins a new

(possibly longer) distinguishable sequence of such type since ρj6= 0.

When parts of λs′ below the Durfee square are inserted by O, they never produce

parts of new sizes when the starting partition is an m-falling partition – we saw above

that parts divisible by m of size up to dm

(λ)m have already appeared due to earlier

insertions. They will instead always form repeated parts of sizes divisible by m. If the

largest part of λs′ below the Durfee square is of size equal to the smallest part above

the square, i.e. dm

(λ), then the last part of Lm,n

(λ) is the part dm

(λ)m and inserting

more of these parts – that is, of indistinct size – simply increases the sizes of all parts of

µ rather than producing a new sequence of consecutive parts. Wherever a new size of

part is inserted elsewhere in the partition, however, a repetition of parts of size mi now

intervenes in the sequence (. . . ,mi+ ρj,mi,mi, . . . ,m(i− 1) + ρ

j+1, . . . ). If ρj

= ρj+1,

then the conjugate of (. . . ,mi+ρj,mi,m(i−1)+ρ

j+1, . . . ) ended (in the visible portion

here) with a sequence of consecutive parts of type (. . . , ρj,m−ρ

j, . . . ); with the insertion

of the additional mi, the sequence is of the same type but is no longer of consecutive

parts and so we have separated one sequence into two. If on the other hand ρj6= ρ

j+1,

then the conjugate of the part mi terminated such a sequence in µ and the addition of

repetitions of mi does not change this fact; but these are precisely the sizes of m-strips

we declined to count in establishing nd(λ).

Surveying the insertion process, we see that we have established one new sequence

of type (c,m − c, . . . ) for each size of ρj, one for each size of m-strip above the Durfee

square except for any strips of size exactly k, and one for each size of strip below the

Durfee square save for those we declined to count. Thus, nd(λ) = n

m(µ).

48

Since conjugation and O are reversible operations well-defined on the entirety of

the relevant sets (Stockhofe shows the latter), the theorem is proved.

It should be noted that the bijection η satisfies all conditions of Theorem 8 except

the last; when residues in ρ differ, the number of sequences of consecutive parts in η(λ),

and even more so the number of appearances of each part, depends in a convolved way

on the values of residues at somewhat symmetric distances from the ends of ρ.

The reversibility of our map tells us that

Corollary 5. The number of m-falling partitions λ of n with given l(λ), dm

(λ), bλ1m c,

and nd(λ), equals the number of partitions µ of n of m-alternating type; and with l

a(µ),

µ1, bl(µ)+m−1

m c, and nm

(µ) related to the statistics of λ as above.

3.2.1 Generating Functions

If Pm

arem-falling partitions (a subset ofm-odd partitions) andDm

are partitions

of m-alternating type (a subset of m-distinct partitions), then we have the identity

∑λ∈Pm

xnd(λ)

yl(λ)

zl(λ)+bλ1

m ctdm∗(λ)

q|λ| =

∑µ∈Dm

xnm(µ)

yla(µ)

zµ1tb l(µ)+m−1

m cq|µ| .

Naturally, we desire to associate to this theorem a generating function identity

more informative than the one described above, and perhaps find some interesting special

cases.

49

Let us begin with the simplest nontrivial extension of previous theorems: consider

a partition λ = (λ1, . . . , λk), with residue-vector ρ = (c1, c1, . . . , c1, c2, c2, . . . , c2), where

c1 > c2. In such a case, the m-modular diagram of Lm,n

(λ) can be decomposed in the

following visual fashion:

Fig. 3.3. The two-residue case.

Here we see two partitions which, by themselves, would be Lm,n

applied to an

(m, ci) partition, and a rectangle of m-weights. Supposing the the lower partition were

the image of such an (m, c1) partition, say α, then it would have b1 parts of residue c1 –

i.e., l(α) – and a number of parts divisible by m equal to bα1m c, so this is the length of the

rectangle. Of course, for (m, c) partitions, l(α)+ bα1m c is exactly the statistic counted by

z in the identity above. The rectangle’s height is either the m-weight of the largest part

in the partition to the right – this is exactly the statistic indexed by n if that partition’s

largest part is divisible by m – or 1 less than this if not.

50

Letting A1 be any given set of (m, ci) partitions with nonempty ρ, the generating

function as given in [16] is:

F1(x, y, z, t, q;m; ci) =

∑λ∈A1

xnd(λ)

yl(λ)

zl(λ)+bλ1

m ctdm∗(λ)

q|λ|

=∑ni≥0

xtni+1

yni+1

z2ni+1

q(ni+1)(mni+ci)

[1

1− yzqmni+ci+

zqm(ni+1)

1− zqm(ni+1)

+xyz

2qm(2ni+1)+ci

(1− zqm(ni+1))(1− yzqmni+ci)

]×

((1− x)zqm; qm

)ni

((1− x)yzqci ; qm

)ni(

zqm; qm)ni

(yzqci ; qm

)ni

.

(3.1)

(The form is here altered slightly from [16] to make combinatorial interpretation

easier.)

In constructing the generating function for multiple residues we need to concern

ourselves with whether the largest part of the lower component partitions are divisible

by m. The four terms in the square brackets in 3.1 determine whether or not the largest

part of the η(λ) is divisible by m, and if so whether this arises from insertion of a distinct

length of m-strip. We break these two cases out for the multiple-residue case since this

datum affects some of the statistics we consider.

For the two-residue case we have constructed our diagram from any (m, c2) par-

tition, and any (m, c1) partition, with an additional m-weight on each part of the latter

equal to the size of dm

(λ) in the preimage of the former. Of the statistics we are examin-

ing in the overall partition, l(µ) and la(µ) are precisely the sums of the relevant statistics

51

for each of the smaller (m, c) partitions, and |µ| is the sum of the two plus that for the

upper rectangle. On the other hand, b l(µ)+m−1m c for the overall partition is 1 less than

the sum of the two partitions’ relevant statistics when the lower partition’s largest part

is not a multiple of m – so we must reduce the weight of t by 1 – and we have one extra

m-chain (of length c1 − c2) in such a case when the first part of the upper partition is

immediately followed by an inserted m-strip – so we must increase the weight of x by 1

in such cases.

Finally, the bijection L1,n Lm,n

interchanges partitions of type

(c2,m− c2, c2,m− c2, . . . , f(c1),m− c1, c1,m− c1, . . . ) with partitions of residue-vector

ρ = (c1, c1, . . . , c1, c2, c2, . . . , c2), so letting A2(c1, c2) now be the set of such partitions

with at least one each of residues c1 and c2 we have that the relevant generating functions

are both

52

F2(x, y, z, t, q;m; c1, c2) =∑

λ∈A2(c1,c2)

xnd(λ)

yl(λ)

zl(λ)+bλ1

m ctdm∗(λ)

q|λ|

=∑n2≥0

x(tyqmn2+c2)n2+1z2n2+1

((1− x)zqm; qm

)n2

((1− x)yzqc2 ; qm

)n2(

zqm; qm)n2

(yzqc2 ; qm

)n2

×[

11− yzqmn2+c2

×

((xyzqmn2+c1

[1

1− yzqmn2+c1+

xzqm(n2+1)

1− zqm(n2+1)+

xyz2q2mn2+1+c1

(1− zqm(n2+1))(1− yzqmn2+c1)

])

+∑n1≥1

xtn1(yqmn1+c1)

n1+1(zqmn2)2n1+1

[1

1− yzqm(n2+n1)+c1+

zqm(n2+n1+1)

1− zqm(n2+n1+1)

+xyz

2q2m(n2+n1)+1+c1

(1− zqm(n2+n1+1))(1− yzqm(n2+n1+c1)

]

×

((1− x)(zqmn2)qm; qm

)n1

((1− x)y(zqmn2)qc1 ; qm

)n1(

(zqmn2)qm; qm)n1

(y(zqmn2)qc1 ; qm

)n1

(1− (1− x)(1− yzqmn2+c1))

+zqm(n2+1) − (1− x)yz2qm(2n2+1)+c2

1− zqm(n2+1)(1− yzqmn2+c2)

∑n1≥0

x(tyqmn1+c1)n1+1

(zqm(n2+1))2n1+1

×

[1 +

(zqm(n2+1))qm(n1+1)

1− (zqm(n2+1))qm(n1+1)+

y(zqm(n2+1))qmn1+c1

1− y(zqm(n2+1))qmn1+c1

+xy(zqm(n2+1))2qm(2n1+1)+c1

(1− (zqm(n2+1))qm(n1+1))(1− y(zqm(n2+1))qmn1+c1)

]

×

((1− x)(zqm(n2+1))qm; qm

)n1

((1− x)y(zqm(n2+1))qc1 ; qm

)n1(

(zqm(n2+1))qm; qm)n1

(y(zqm(n2+1))qc1 ; qm

)n1

. (3.2)

53

We can repeat this nesting process to get the generating functions for 3 or more

types of residues, up to the generating function for those m-falling partitions in which

every residue 1, . . . ,m− 1 appears.

3.2.2 Appearance of Descents

The situation quickly becomes more complex when λ is permitted descents in

the residue-vector, mostly in discussing the number of chains in µ. For the remaining

statistics, slight adjustments need to be made to the equalities but the relations remain

similarly structured.

If we neglect consideration of chains, then the same theorem can be established

using either the generalized bijection η I give above, or using L1,n Lm,n. Since we

have already done most of the examination we will need for the latter, and with a view

toward later establishment of a better identity for chains using what appears to be a

more useful tool, we will continue to perform our analysis using L1,n Lm,n.

Denote by fλ

the number of descents in the residue-vector ρ of λ, that is, the

number of ρi

such that ρi< ρ

i+1. Then:

Theorem 9. Let λ be a partition of n > 0 into parts not divisible by m, λ = mρ+mλ

s.

Then µ = L1,n Lm,n(λ) is a partition of n into parts appearing less than m times in

which

• l(λ) = la(µ);

• µ1 = l(λ) + bλ1m c − fλ; and

• b l(µ)+m−1m c = f

λ+ 1 + #

((λs

)′)i> i+ f

λ.

54

Furthermore, L1,nLm,n exchanges all m-odd and m-distinct partitions with these statis-

tics.

(It must be noted that the second clause of the list appears as – or, rather, can

viewed as a straightforward rewrite of – Theorem 3.8 (ii) in [13].) A statement similar

to those in previous theorems regarding the number of chains in µ can be constructed,

but inelegantly, requiring knowledge of the positions of the descents in ρ in comparison

to the lengths of m-strips, and the presence of descent-patterns in the residue-vector of λ

given by (. . . , ρj−1 +mb, ρ

j+m(b− 1), ρ

j+1 +m(b− 1), . . . ), where ρj−1 = ρ

j+1 < ρj,

or similarly with ρj+1 > ρ

j.

Proof. The first claims are the same as in the case without descents. Once again we

note that L1,n Lm,n interchanges m-odd partitions and m-distinct partitions, so µ is

m-distinct. Lm,n

preserves ρ, since Lm,n

(mρ +mλs) = m

ρOmλs′, and hence l

a(L1,n

lm,n

(λ) = l(mρ) = l(λ).

We begin analysis of the remaining claims by noting that |mρ1|m = fλ. (Indeed,

the m-weights in mρ are essentially the dual of the greater index of ρ considered as

a permutation of a multiset in [1, . . . ,m − 1]: if ρ is of length k, then∑i|mρ

i|m

=

k ∗ fλ− ind(ρ).)

Since µ1 is simply the number of parts in Lm,n

(λ), we note that this is the length

of ρ – i.e., the length of λ – plus the number of strips inserted by O, i.e. (λs)1. We have

bλ1m c = |mρ1|m+(λ

s)1

= fλ+(λ

s)1, so (λ

s)1

= bλ1m c−fλ and thus µ1 = l(λ)+bλ1

m c−fλ.

If the longest m-strip of λ is of length fλ

or less, say i, then where O inserts

the part mi is between the two parts of ρ that constitute the i-th descent in ascending

55

order of size: i.e., as (. . . , ρj

+ mi,mi, ρj+1 + m(i − 1), . . . ), where ρ

j< ρ

j+1. If the

strip is of length exactly fλ

+ 1, it is inserted as the new largest part of Lm,n

(λ) :=

(m(fλ

+ 1), ρ1 +mfλ, . . . ). If finally the length of the first strip is greater than f

λ+ 1,

again say i, thenm will be added to the first j parts ofmρ and the remaining weight of the

strip will be inserted as a part of size mb: forming (. . . , ρj+mb,mb, ρ

j+1+m(b−1), . . . ),

where ρj≥ ρ

j+1 and i = j + b. (While it is possible that i = j + b for a consecutive

sequence of mρj

with |mρj|m

= b with decreasing b and increasing j, this only happens

if each of the ρj

involved are the larger sides of a descent. In such a case mb can only be

inserted at the beginning of the sequence, as (. . . ,m(b+1), ρj+mb, . . . ), since insertion

anywhere within the sequence would produce two parts (. . . , ρj+mb,m(b− 1), . . . ) that

differed by more than m.) For the various possible cases, see illustration, next page.

For the second and further m-strips, we repeat the same analysis, bearing in mind

the new m-weight of the largest part of our intermediate partition. If the first m-strip

were of length greater than fλ

+ 1, then this new m-weight is now fλ

+ 1. In order to

increase this weight again the length of the second m-strip must be at least fλ

+ 2, etc.

If there is no strip((λs

)′)i

of length exactly i + fλ, then the largest part of

mρOλ

swill be ρ1 +m(f

λ+ #

((λs

)′)i> i+ f

λ). If on the other hand there is some

i such that((λs

)′)i

= i + fλ, then the largest part of mρOλ

swill be m(i + f

λ) =

m(fλ

+1+#((λs

)′)i> i+f

λ. Thus b l(µ)+m−1

m c = fλ

+1+#((λs

)′)i> i+f

λ.

The reversible nature of L1,n Lm,n for any partition into parts appearing fewer

than m times tells us that we have matched the entirety of both sets under consideration,

with the statistics related as shown above.

56

Fig. 3.4. Cases of O with descents in ρ.

57

This proof provides the theorem cited at the top of the chapter, with the additional

statistics we promised.

We mentioned that counting m-chains – sequences of parts differing by at most 1,

in which the largest part appears c times, the next largest part appears m− c times, the

next appears c times, etc. – was complicated by the possibility of descent-patterns in the

residue-vector of λ given by (. . . , ρj−1 +mb, ρ

j+m(b− 1), ρ

j+1 +m(b− 1), . . . ), where

ρj−1 = ρ

j+1 < ρj. To see why this is the case, consider λ = (5, 4, 2) with modulus

m = 3. Since λ = mρ, L3,7(λ) = λ and so L1,7 L3,7((5, 4, 2)) = (5, 4, 2)′ = (3, 3, 2, 2, 1).

There is one 3-chain of type (2) and one 3-chain of type (2, 3− 2). The latter, however,

arises not from insertion of a part divisible by 3 (which is how all such m-chains arise in

the m-falling case) but from the simple oscillation of residue differences.

There are at least two possible responses to this problem. One is to take it

as a challenge, and embark upon the study of its ramifications in the context of the

combinatorial theory of word-avoiding permutations of multisets.

This we cheerfully propose for some future date.

The other is to consider a simpler class of partitions in which such patterns do

not occur. In analogy to the m-falling partitions defined earlier, we now consider the

other direction to generalize (m, c) partitions, in which ρ = (c1, c1, . . . , c2, c2, . . . ) with

c1 < c2 < . . . . These we call, of course, m-rising partitions.

In a manner similar to our construction for m-falling partitions, we can construct

a generating function for m-rising partitions by concatenating a sequence of the images of

(m, ci) partitions for rising c

i. Again, most of the statistics being considered are additive.

The considerations for summing the number of chains are exactly the same, dependent

58

on whether the largest part of the (m, c2) partition is divisible by m or not, and whether

the first part of the (m, c1) partition is followed by a part divisible by m. Interestingly,

one facet of the generating function is slightly simpler: regardless of whether or not the

largest part of the (m, c2) partition is mi, the height of the rectangle in the upper left

is constant at m(n2 + 1), for when a descent occurs between c1 and c2, an additional m

must appear.

We must also generalize our statistic nd(λ) completely to cover the case when λ

is an m-rising partition, with descents.

Definition 14. With λ an m-falling or m-rising partition containing fλ

descents, call

nd(λ) the number of distinct kinds of residues in ρ plus the number of distinct sizes of

m-strips in λs, not counting any m-strip of length k+ f

λ(i.e., if there are any descents

at all it does not matter if the smallest part of λ is greater than m), nor any strips of

size i+ b where there exists a strip such that(λs′)i− (i+ b) is exactly the index of the

last appearance in mρ of a ρ

j+mb of a given size.

When λ is an m-falling partition, then a = 0, and b = 0 for all j, since there are

no descents.

We also need to generalize dm∗(λ) for m-rising partitions with f

λdescents to be

fλ

+ 1 + #((λs

)′)i> i+ f

λ. Like before, when f

λ= 0 this definition degenerates to

the previous.

With these generalized definitions in hand we can state a theorem including an

equivalence of nd

and nc:

59

Theorem 10. Let λ be an m-rising partition of n > 0, λ = mρ + mλ

s,

ρ = (c1, c1, . . . , c2, c2, . . . , ck) with 0 < c1 < c2 < · · · < ck< m. Then µ = L1,nLm,n(λ)

is a partition of n of m′-alternating type;

• nd(λ) = n

c(µ);

• l(λ) = la(µ);

• µ1 = l(λ) + bλ1m c − fλ; and

• b l(µ)+m−1m c = f

λ+ 1 + #

((λs

)′)i> i+ f

λ.

Furthermore, L1,n Lm,n exchanges all partitions of such types with these statistics.

Now, let C2(c1, c2), c1 < c2, be the set of m-rising partitions of residue-vector

ρ = (c1, c1, . . . , c2, c2, . . . ) with c1 < c2, and each of c1 and c2 appearing at least

once, and let D2(c1, c2) be those partitions into less than m parts of type (c2,m −

c2, . . . , g(c1),m− c1, c1,m− c1, . . . ), where g(c1) is m+ c1 − c2 if the previous entry is

c2, and c1 if the previous entry is m−c2. The preceding arguments give us the generating

function

60

∑λ∈C2(c1,c2)

xnd(λ)

yl(λ)

zl(λ)+bλ1

m ctdm∗(λ)

q|λ| =

∑µ∈D2(c1,c2)

xnc(µ)

yla(µ)

zµ1tb l(µ)+m−1

m cq|µ|

=∑n2≥0

x(tyqmn2+c2)n2+1z2n2+1

((1− x)zqm; qm

)n2

((1− x)yzqc2 ; qm

)n2(

zqm; qm)n2

(yzqc2 ; qm

)n2

×[

11− yzqmn2+c2

((xyzq

m(n2+1)+c1

×

[1

1− yzqm(n2+1)+c1+

xzqm(n2+2)

1− zqm(n2+2)+

xyz2qm(2n2+3)+c1

(1− zqm(n2+2))(1− yzqm(n2+1)+c1)

])

+∑n1≥1

xtn1(yqmn1+c1)

n1+1(zqm(n2+1))

2n1+1[

1

1− yzqm(n2+n1+1)+c1

+zqm(n2+n1+2)

1− zqm(n2+n1+2)+

xyz2q2m(n2+n1)+3+c1

(1− zqm(n2+n1+2))(1− yzqm(n2+n1+1+c1)

]

×

((1− x)zqm(n2+1)

qm; qm

)n1

((1− x)yzqm(n2+1)

qc1 ; qm

)n1(

zqm(n2+1)qm; qm)n1

(yzqm(n2+1)qc1 ; qm

)n1

(1− (1− x)(1− yzqm(n2+1)+c1))

+zqm(n2+1) − (1− x)yz2qm(2n2+1)+c2

1− zqm(n2+1)(1− yzqmn2+c2)

∑n1≥0

x(tyqmn1+c1)n1+1

(zqm(n2+1))2n1+1

×

[1 +

(zqm(n2+1))qm(n1+1)

1− (zqm(n2+1))qm(n1+1)+

y(zqm(n2+1))qmn1+c1

1− y(zqm(n2+1))qmn1+c1

+xy(zqm(n2+1))2qm(2n1+1)+c1

(1− (zqm(n2+1))qm(n1+1))(1− y(zqm(n2+1))qmn1+c1)

]

×

((1− x)zqm(n2+1)

qm; qm

)n1

((1− x)yzqm(n2+1)

qc1 ; qm

)n1(

zqm(n2+1)qm; qm)n1

(yzqm(n2+1)qc1 ; qm

)n1

. (3.3)

For completeness’ sake we will close this chapter by mentioning that, while we

have not delved into the structure or operation of L1,nLm,n for general partitions (doing

61

so with sufficient detail to prove the relevant claims would require another chapter of

explication), the bijection functions perfectly well on any partition and can be used to

relate the statistics of λ and its image µ in the following fashion. Fixing a modulus m

and letting P rk,l

(n) be the union over all ρ of length r of Pρk,l

(n), that is, all partitions

of n with k parts divisible by m and l m-strips, we have

Theorem 11. Let λ ∈ Pr

k,l(n) with its mρ possessing f

λdescents. Then µ = L1,n

Lm,n

(λ) is a partition of n in which

• l(λ)− k = la(µ);

• µ1 = l(λ)− k + l; and

• b l(µ)+m−1m c = f

λ+ 1 + k + #

((λs

)′)i> i+ f

λ.

Furthermore, L1,n Lm,n exchanges all partitions with these statistics.

62

Appendix A

Notation

Some of the more commonly used specialized notation in this thesis:

p(n) The number of partitions of n.

N(b, p, n) Number of partitions of n with rank ≡ b (mod p).

N(m,n) Number of partitions of n with rank m.

ra,b

(q; p; d)∑n≥0 q

n(N(a, p, n)−N(b, p, n)).

R1(z; q)∑∞z=−∞

∑n≥0N(m,n)zmqn.

ζp

e2πip

Dl(n) Number of l-marked Durfee symbols of n.

Dl(m1, . . . ,ml;n) ” with ith ranks all m

i.

NFl(m,n) ” with full rank m.

NFl(b, p, n) ” with full rank ≡ b (mod p).

Pρ

k,l(n) Partitions of n with residue-vector ρ,

k parts divisible by m, and l m-tags.

mρ The unique partition in Pρ

0,0.

fλ

The number of descents in the residue-vector ρ(λ).

63

Appendix B

Translation: Stockhofe’s Thesis

This appendix is an English translation of the German original of Dieter Stock-

hofe’s thesis, [13], that I produced in the course of studying for this work. It is provided

in support of the material of chapter 3, and also as a general service to the Anglophone

mathematical community. I found the tools therein useful and interesting, and have not

made full use of all the variations elaborated upon therein for this thesis.

Caveat lector: I am not a professional translator. For the math (especially as used

in this thesis), I vouch. For the German, I will not be seeking any awards. Being an

appendix to a larger work, the chapters of Stockhofe’s thesis are repurposed as sections,

and the numbering of equations has therefore altered significantly from the original.

64

B.1 Foreword

The theory of partitions is understood to be a subdivision of additive number

theory. The first formulated questions leading to the theory had already been brought

up deep in the Middle Ages; however, they were proved valid by L. Euler. He supplied

many fundamental contributions to the theory, before Cauchy, Jacobi, Sylvester, Hardy,

Ramanujan, Rademacher, and many other mathematicians would further expand it. It

was soon shown that partitions also frequently play a role in mathematics concerning the

parametrization and classification of mathematical objects, for example of finite abelian

groups or the irreducible representations of symmetric and complete linear groups.

The investigation of the problem of how many ways a positive whole number can

be written as a sum of positive whole numbers,

n = n1 + · · ·+ nw, w ∈ N

(two sums being considered the same provided that they are only different in the order

of their summands) led to the idea of a partition of a natural number and gave occasion

for the following definition.

Definition B.1. A partition α is a finite sequence of positive whole numbers

α := (α1, . . . , αw), w ∈ N

such that

α1 ≥ α2 ≥ · · · ≥ αw .

65

The αiare the parts of α, and w is the length of the partition α. α is called a partition

of n when

|α| :=w∑i=1

αi= n .

P (n) denotes the set of all partitions of n. We additionally set P (0) := 0.

An important object of this theory is the enumeration of subsets T ⊆ P (n), where

elements are distinguished by certain properties, and the comparison of such subsets

regarding their cardinalities. The proof methods are in part of the combinatoric type, in

part of the analytic type, and primarily analytic where the investigation of generating

functions is concerned.

In this work the combinatorial aspect will be in the foreground. For all natural

numbers n, q ∈ N, we will construct bijective transformations Lq,n

of P (n) in Chapter

2, and then in Chapters 3 and 4 will derive some enumeration theorems. Thus follows,

for example, the following counting theorem for partitions:

Denote by |m|q

the largest whole number divisible by q which is less than or equal

to m, and α′ the partition conjugate to α; then it holds that for all k, l ∈ N0 the number

of all partitions α of n with

ψq(α) :=

∣∣∣α1 − α2

∣∣∣q

+∣∣∣α2 − α3

∣∣∣q

+ · · · = k

and

χq(α) := α

′q− α′

q+1+ α′2q− α′

2q+1± · · · = l

is equal to the number of all partitions β of n with χq(β) = k and ψ

q(β) = l.

66

The case q = 2 deserves particular interest, since well-known identities of Euler,

Sylvester, and Fine follow from that special case.

Finally, in Chapter 5 we will state a counting formula for the fixed points of

Lq,n

, and in Chapter 6 it will be shown that the bijections L1,n, . . . , Ln−1,n generate

the entire symmetric group of P (n). Since the proof is constructive, one can for any

bijective transformation f of P (n) give a sequence of natural numbers q1, . . . , qr with

r ∈ N, 1 ≤ qi≤ n− 1, so that f can be written in the form

f = Lq1,n· · · · · L

qr,n.

I would like to cordially thank Herrn Prof. Dr. A. Kerber for many worthy sug-

gestions and indications for this work.

67

B.2 Notation

N0 N ∪ 0

Nn

1, 2, . . . , n

|M | number of elements of the setM

a | b a is a divisor of b

idM

the identity transformation of the setM

U ≤ V U is a subgroup ofV

U w V U is isomorphic toV (as a group)

SM

symmetric group of the setM

Sn

SNn

Dn

dihedral group of order 2n

<M> the subgroup generated byM

B.3 Graphical Representations of Partitions: the q-modular diagram

In a mathematical theory, it often happens that one would like to have concrete

representations of applied concepts. Serving as an important expedient for graphical

representation of partitions is the so-called Young- or Ferrers-diagram:

68

Definition B.2. If α = (α1, . . . , αw) is a partition, then the set of all lattice points

(i, j)|i, j ∈ Z, 1 ≤ i ≤ w, 1 ≤ j ≤ αi

is called the Young Diagram of α.

In connection with the graphical representation of the Young diagram we strike

– in the converse of the customary convention – the following agreement: the first co-

ordinate i grows from West to East, and the second coordinate j grows from North to

South. For each lattice-point in the Young diagram of α we write, for reasons that will

be obvious later, a 1, and denote the pattern of all 1s (which we shall call the 1-Diagram

of α ) with [α]1 or [α].

Example B.3. For α := (5, 4, 4, 2) =: (5, 42, 2) we have

[α]1 =

1 1 1 1

1 1 1 1

1 1 1

1 1 1

1

.

The use of the Young diagram makes the following definition more plausible:

Definition B.4. If α =(α1, . . . , αw

)is a partition, then by

α′i:=∣∣∣j ∈ N|α

j≥ i

∣∣∣

69

we define the partition α′ :=

(α′1, . . . , α

′α1

)to be the conjugate partition to α.

Obviously one gets that the 1-diagram belonging to α′,[α′]

1, is just [α]1 reflected

across its main diagonal (the NW-SE axis). So one has for the example

[(5, 4, 4, 2)′

]1

=

1 1 1 1 1

1 1 1 1

1 1 1 1

1 1

.

With this it is immediately clear that the transformation C with C(α) := α′,

because C2 = id, is a bijection of the set of all partitions. C is the conjugating transfor-

mation. Partitions for which α = α′ holds, we call self-conjugate.

The representation of partitions by means of diagrams we shall now generalize

from Example B.3. In the following it will always hold that q ∈ N is a fixed natural

number. If m ∈ N0, we will understand by the q-length |m|q, the greatest whole number

that is less than or equal to mq . We may then write m in the form

B.5. m = |m|qq + r

q(m),

whereby rq(m) ∈ Z, by means of the stipulation 0 ≤ r

q(m) < q, is guaranteed to be

unique. rq(m) is called the q-residue or also for short the residue of m.

As in B.5 one breaks up now all parts αi

of α =(α1, . . . , αw

)and makes for

it a rectangular number-pattern consisting of w columns, which end at rising levels,

wherein one writes in the ith column the residue rq

(αi

), if in fact r

q

(αi

)> 0 holds,

70

and thereunder enters the number q∣∣∣αi

∣∣∣q

times (see Example B.7). We will call these

number patterns q-modular diagrams of α, or q-diagrams for short, and denote them

with [α]q. If α is a partition of n, then we also say that [α]

qis a q-diagram of n.

An entry i of [α]q

is called a (q-)residue unit in the case that i < q; otherwise it

is a q-unit. For the number of all q-units of [α]q, the so-called q-weight, we write |α|

q.

The residue units of [α]q

will – numbered consecutively from left to right – moreover

comprise the (q-)residue-vector

B.6. ρ := ρα :=

(ρ1, . . . , ρr

).

When [α]q

contains no residue units, then we set r := 0 and ρα := ∅.

If the units in the columns are arranged in such a way that the residue unit

(insofar as it is available, i.e. 6= 0) forms the lower end of a column in each case, then a

diagram is developed that we will denote with [α]∗q.

71

By way of illustration I provide the following example:

Example B.7. α :=(42, 36, 24, 20, 12, 102

, 8, 7, 5)

; q := 5

Fig. B.1. Part (i)

ρα = (2, 1, 4, 2, 3, 2)

72

Fig. B.2. Part (ii)

A part αiof α is called (q-)singular ((q-)regular), in the case where q | α

i(q - α

i).

(The q-singular parts of α belong, in [α]q

and/or [α]∗q, to exactly those columns which

hold no residue units.) A partition α is called (q-)singular ((q-)regular) in the case where

every part αi

is (q-singular) ((q-)regular). If α is a q-singular partition, then one can

write α in the form

B.8. α := qβ :=(qβ1, . . . , qβw

),

with β :=(β1, . . . , βw

)∈ P .

The following idea is only to be seen in connection with the q-diagram [α]q

(and

not with [α]∗q

): a q-unit xiin [α]

qis called a (q-)edge unit if the q-diagram that develops

when one removes xi

and all q-units located under xi

in the same column is again a q-

diagram of a partition β (these are marked in Example B.7 Part (i) by squares). 1

1Translator’s note: Stockhofe appears to consider a column to exist, and represent 0, if all ofits entries are removed. This may cause a partition to have trailing zeros which are not countedas parts.

73

The edge units of [α]q

– numbered consecutively from the top down – are x1, . . . , xl.

To each xi

then belongs a subset Si

of [α]q, a so-called (q-)strip which is built up as

follows: Sicontains the lowest q-unit in each column left of x

inot lying in S

i+1∪· · ·∪Sl,

and xi

itself (in Example B.7 , thus, exactly those q-units which are on the line drawn

through xi).

A partition α =(α1, . . . , αw

)is called (q)-flat if, for 1 ≤ i ≤ w, α

i− α

i+1 < q

holds (one sets αw+1 := 0). α is thus flat precisely if [α]

qpossesses no edge units.

Thus, when all strips Sl, . . . , S1 are removed from [α]

q, the q-diagram finally remaining

is a flat partition αf. α

fis the (q-)flat portion of α. In Example B.7 this is α

f=(

17, 16, 14, 10, 7, 52, 3, 2

).

If the regular columns of [α]q

(columns which correspond to regular parts of α )

are put together into a q-diagram[αr

]q

and thereafter all strips from[αr

]q

are removed,

there then remains a (q-)residue diagram whose associated partition – we will denote it

with mα – is regular and flat. As one sees easily, a partition which is both regular and

flat is clearly fixed by indication of the residue-vector ρ. We write then also mρ with

ρ := ρα for the residue diagram of α.

Example B.9. (i) For q = 1 each partition α ∈ P is singular and it follows that

[mα]1

=[mα]

= ∅ .

(ii) For q = 2 all partitions which are flat and regular are of the form α =(1r)

with

r ∈ N. It follows that[mα]

2 =[1r]2 if r is the number of all odd parts of α.

(iii) Let α and q be as in Example B.7. Then

74

[mα]q

=

2 1 4 2 3 2

5 5 5 5

5 5

.

Note: Given [α]q

one does not generally get the diagram[mα]q

by first removing all the

strips from [α]q

and then from the diagram that remains omitting the singular columns.

For example, regard the diagram

[α] =1 q 1

q

,

if q ≥ 2 holds. Here the diagram1 1

q

remains, whereas[mα]q

= 1 1 .

Set r ∈ N0 and ρ ∈ Nrq−1

. For all k, l ∈ N0 we define

Definition B.10. Pρk l

:= α ∈ P |ρα = ρ, [α]qhas k q-singular columns and l

q-edge units

and

Definition B.11. P rk l

:=⋃

ρ∈Nrq−1

Pρ

k l.

Pr

k lis thus the set of all partitions of r regular and k singular parts with

∣∣∣α1 − α2

∣∣∣q

+∣∣∣α2 − α3

∣∣∣q

+ · · · = l.

75

For example, the partition α in Example B.7 is an element of P 6

4 5.

Remark on the notation: In the following sections subsets of P are frequently regarded

as combinations of sets P rk l

with r, k, l allowed to be written as elements of N0. In order

to have a clear and easy-to-read notation here, the following abbreviations are to be

agreed upon:

(i) We write P≤rk l

in place of⋃

0≤i≤rPi

k land set P

k l:=

⋃0≤i

Pi

k l. Similar conventions apply

to the two lower indices. Thus for example P≤r≤K l

=⋃

0≤i≤r0≤j≤k

Pi

j l, and P· 0 =

⋃0≤i,j

Pi

j 0.

(The · in P· 0 is to suggest that the first of the two lower indices is omitted.)

(ii) For all subsets T of P it will be that T (n) := T⋂P (n).

From the past definitions some direct consequences now result:

Theorem 12. The partition α is (q-)flat exactly if α′ has at most q − 1 equal parts.

Theorem 13. α is regular exactly if α is an element of P0 ·; α is flat in exactly the case

where α lies in P· 0.

Theorem 14. The set Pρ0 0

is one-element, namely: Pρ0 0

= mρ. Therefore α lies in

Pρ

0 0exactly when α = m

α holds.

Theorem 15. Pρ(n) is nonempty exactly when∣∣mρ∣∣ ≤ n and n ≡

∣∣mρ∣∣ mod q. From

Pρ(n) 6= ∅ it follows in particular that

∑r

i=1ρi≡ n mod q.2

Theorem 16. For the q-length of the parts of mρ, it holds that

∣∣∣∣mρi

∣∣∣∣q

=∣∣∣j|i ≤ j ≤ r − 1, ρ

j< ρ

j+1∣∣∣ .

2When p is a partition of n, |p| = n.

76

Theorem 17. From Theorem 16 it follows, for the q-weight of mρ:

∣∣∣mρ∣∣∣q

=r−1∑i=1

ρi<ρi+1

i .

Theorem 18. The q-weight of mρ is – for fixed r – maximal exactly when the case

ρ1 < · · · < ρr

holds; in this case it results that

∣∣∣mρ∣∣∣q

=r−1∑i=1

i =(r

2

).

Theorem 19. Set α′ =(α′1, . . . , α

′w

)to be the conjugate partition of α. Then α is an

element of Pk · exactly when

α′q− α′

q+1+ α′2q− α′

2q+1± · · · = k

holds. (One represents α by the diagram [α]∗q.)

In the next section, q-diagrams which are distinguished by certain characteristics

are joined to new q-diagrams. In addition we define for partitions α =(α1, . . . , αw

)and β =

(β1, . . . , βv

)the sum of α and β by the appointment

(α+ β)i:= α

i+ β

i

for all 1 ≤ i ≤ maxw, v. (If v < w, then one sets βv+1 := · · · := β

w:= 0; one proceeds

analogously in the case w < v.) Furthermore, the union α⋃β is the partition which

77

develops if one collects the parts of α with those of β and rearranges them by size. Here

it is to be still noticed that (α⋃β)′ = α

′ + β′ applies.

If for example α := (3, 2) and β := (4, 2, 1), then α + β = (7, 4, 1) and α⋃β =

(4, 3, 2, 2, 1).

If the q-units of each strip of [α]q

are written next to each other in a row, and

then all left justified by size among themselves, then there develops a singular diagram[qαs

]q, so that

B.12. α = αf

+ qαs

applies. Obviously B.12 is the only possible representation of α as the sum of a flat and

a singular partition. Then α is an element of Pρ· l

exactly if αf

is an element of Pρ· 0

and

qαs

is an element of P 0

· l.

B.4 Construction of a Bijective Transformation Lq on the Set of All

Partitions

In this chapter there is constructed – for fixed q – a transformation Lq

of P ,

which can be understood as a generalization of the conjugating transformation defined

in Chapter B.3.

B.4.1 Characteristics of q-flat Partitions

We begin with the investigation of flat diagrams, in particular on the possibility of

inserting singular columns and “angles” in such a way that a flat diagram again results.

In a flat partition α =(α1, . . . , αw

)set i1 < · · · < i

rthe indices of the regular

parts. We set now (in order to avoid later unnecessary definitions by cases) i0 := 0,

78

ρ0 := 03 and αi0

:= q

(∣∣∣∣αi1∣∣∣∣q

+ 1

). Obviously the difference between the q-lengths of

αij−1

and αij

is, for 1 ≤ j ≤ r, either 0 or 1. It is, as one may easily consider, 1 exactly

if the residue of αij−1

is smaller than the residue of αij

, or if there is a singular part

between αij−1

and αij

. We have thus for 1 ≤ j ≤ r:

B.13.

∣∣∣∣αij−1

∣∣∣∣q

=

∣∣∣∣αij∣∣∣∣q

+ 1 if ρj−1 < ρ

jor i

j> i

j−1 + 1;∣∣∣∣αij∣∣∣∣q, otherwise.

In light of B.13 we now want to number consecutively the indices of the regular

parts of α, so that k1 > k2 > · · · > kt

are the indices of the regular parts of α for

which∣∣∣∣αij−1

∣∣∣∣q

=∣∣∣∣αij

∣∣∣∣q

+ 1 holds (in Example B.14 these are marked by “←” ), while

kt+1 < k

t+2 < · · · < kr

should be the indices of the remaining regular parts of α

(marked in B.14 by “→”). Then we have:

Theorem 20. (i)∣∣∣∣αkj

∣∣∣∣q−∣∣∣∣αkj−1

∣∣∣∣q

= 1 for 2 ≤ j ≤ t, and

(ii)

∣∣∣∣αkj−1

∣∣∣∣q−∣∣∣∣αkj

∣∣∣∣q

=∣∣∣i|1 ≤ i ≤ t, k

j−1 < ki< k

j∣∣∣

= kj− k

j−1 − 1−∣∣∣l|k

j−1 < l < kj, q | α

l∣∣∣

for t < j ≤ r.

3Translator’s note: Residues are ρ .

79

The difference in 20 (ii) is thus equal to the number of all regular parts between

kj−1 and the k

j-th part.

Example B.14. In [α]5 :=

k5 k6 k4 k3 k2 k7 k8 k1

→ → →

← ← ← ← ←

5 3 2 4 5 2 4 1 1 5 5 1

5 5 5 5 5 5 5 5 5

5 5 5 5 5 5

5 5 5 5

5 5 5

we have k1 = 12, k2 = 7, k3 = 6, k4 = 4, k5 = 2 = kt

and k6 = 3, k7 = 8,

k8 = 9.

Now, if the diagram [α]q

is to be extended, by inclusion of a suitable number

of q-units, to a diagram [α]q, in such a way that [α] is likewise flat but has one more

singular column than [α]q, then there are exactly r possibilities:

Theorem 21. (i) If 1 ≤ j ≤ t, one can insert a singular column direectly to the left of

the kj-th column. This column obviously consists of exactly

∣∣∣∣αkj∣∣∣∣q

+ 1 =: fα(j) q-units.

(ii) If t < j ≤ r, one can insert a singular column directly to the left of the kj-

th column, if in addition all columns to the left of the kj-th column are increased by a

q-unit. In total then∣∣∣∣αkj

∣∣∣∣q

+ kj

:= fα(j) q-units are necessary.

In the future we shall name a pattern of q-units inserted in the manner of 21 (ii)

a (q-)angle.

80

For example, one develops, from [α]5 above, the diagram

Fig. B.3. A column inserted.

by inserting a singular column to the left of αk4

.

If however we insert a singular column to the left of αk7

in accordance with 21

(ii), then there results the diagram B.4.

We now concern ourselves with examining the function fα from Theorem 21.

fα is a strictly monotonic increasing function of N

rin N, since using Theorem 20 one

concludes inductively:

Theorem 22.

(i) fα(j) = j for 1 ≤ j ≤ t

(ii) fα(j) = j +

∣∣∣l|l < kj, q | α

l∣∣∣ for t < j ≤ r

= j + the number of singular columns

left of the kj-th column in [α]

q.

81

Fig. B.4. An angle inserted.

In Example B.14 we have fα(j) = j for 1 ≤ j ≤ 5, in particular fα(4) = 4. On

the other hand, fα(6) = 7, fα(7) = 9, and fα(8) = 10.

If [α]q

is a flat diagram, and [α]q

is that diagram extended by a singular column

or angle inserted left of the αkj

-th column according to the method in Theorem 21, then

it follows from Theorem 22 that

Theorem 23. f α(i) = fα(i− 1) + 1 , when one sets f

α(0) = 0 .

B.4.2 A Bijective Transformation from P0

· ≤rto P

ρ

· 0

Crucial to the construction of Lq

is a bijective transformation σρ(∀r ∈ N0, ρ ∈ Nr

q−1

)of the set of partitions

P0

· ≤r= γ|γ = qβ =

(qβ1, . . . , qβk

), k ∈ N0, β1 ≤ r

into the set of flat partitions Pρ· 0

.

82

Let ρ ∈ Nrq−1

and the partition mρ be given. From m

ρ and qβ ∈ P 0

· ≤rwe build

up a partition we will denote by mρ 5 qβ. One constructs, proceeding from

[mρ]q, a

sequence of flat q-diagrams,

B.15.([mρ]q

=:[βρ,0]q,[βρ,1]q, . . . ,

[βρ,k]q

),

in such a manner that for all 1 ≤ i ≤ k

B.16.[βρ,i]q

develops from[βρ,i−1

]q

by inserting βiq-units in accordance with The-

orem 21

and sets then

B.17. mρ 5 qβ := βρ,k =: σρ (qβ) .

Since fα is a strictly monotonically increasing function which is injective for all

α ∈ Pρ

· 0, there is for each diagram

[βρ,i−1

]q, 1 ≤ i ≤ k, given in B.15, exactly one

diagram[βρ,i]q

which meets condition B.16.

The existence of[βρ,0]q

is seen as follows: for all 1 ≤ i ≤ r we have from Theorem

22

fmρ

(i) = i .

Since we supposed βi≤ r, there can be (by Theorem 21) inserted into m

ρβ1

q-units, so that a new singular column and/or a new angle develops (left of the kβ1

-st

column). From Theorem 23 it follows now that for all 1 ≤ i ≤ β1:

fβρ,1

(i) = fmρ

(i− 1) + 1 = i .

83

Since β2 ≤ β1, one can insert β2 q-units into[βρ,1]q, so that a new singular col-

umn and/or a new angle (left of the kβ2

-nd column) develops. One then has, inductively

with Theorem 23,

fβρ,j

(i) = i for all i ≤ βj,

and can conclude the existence of[βρ,j+1

]q

from the existence of[βρ,j]q.

Completely similarly one can show that to each partition α ∈ Pρk 0

one can con-

struct a sequence of q-diagrams

([α]

q=:[γρ,k]q,[γρ,k−1]

q, . . . ,

[γρ,0]

q=:[mρ]q

), B.15∗

so that for all 1 ≤ i ≤ k

[γβ,i]qdevelops from

[γβ,i−1]

qby inserting a column or an angle of γ

iq-units, and

(qγ1, . . . , qγk

)=: qγ is an element ofP 0

· ≤r.

B.16∗

The transformation

σρ : Pρ· 0→ P

0

· ≤rdefined by

σρ (α) := qγ B.17∗

is obviously the inverse mapping to σρ from B.17. It follows that σρ is bijective.

84

We have thus shown:

Theorem 24. For all r ∈ N0, ρ ∈ N0

q−1the transformation σ

ρ : P 0

· ≤r→ P

ρ

· 0

, defined by σρ (qβ) = m

ρ 5 qβ , is bijective, and one has for all k ∈ N0, qβ ∈ P0

· ≤r:

(i)∣∣σρ (qβ)

∣∣q

=∣∣mρ∣∣

q+ |β|

(ii) σρ(P

0

k≤r

)= P

ρ

k 0.

Example B.18. Set q = 5, ρ = (1, 4, 3, 1, 1, 4, 2, 1) and β = (8, 7, 7, 5, 1). Then it is that

[mρ]q

=[βρ,0]

q=

1 4 3 1 1 4 2 1

q q q q q

q

[βρ,1]

q=

1 4 3 1 1 4 2 q 1

q q q q q q q

q q q q q

q

[βρ,2]

q=

1 4 3 1 q 1 4 2 q 1

q q q q q q q q

q q q q q q

q q q q

85

[βρ,3]

q=

1 4 3 q 1 q 1 4 2 q 1

q q q q q q q q q

q q q q q q q

q q q q q

q q q

q

[βρ,4]

q=

1 q 4 3 q 1 q 1 4 2 q 1

q q q q q q q q q q

q q q q q q q q

q q q q q q

q q q q

q

[βρ,5]

q=

1 q 4 3 q 1 q 1 4 2 q q 1

q q q q q q q q q q

q q q q q q q q

q q q q q q

q q q q

q

86

For q = 5, for example, it is thus that

σρ (qβ) = m

ρ 5 qβ =(26, 25, 24, 23, 20, 16, 15, 11, 9, 7, 52

, 1)

.

Corollary 6. From the proof of Theorem 24 it follows, that during the construction of

mρ 5 qβ , exactly those columns of [qβ]

qbecome angles, for which it holds that the

βi− i >

∣∣∣∣mρ1∣∣∣∣q

. From this one also sees that the first column of[mρ 5 qβ

]then

becomes singular exactly when an i ∈ N exists with βi− i =

∣∣∣∣mρ1∣∣∣∣q

.

B.4.3 A Bijective Transformation from P to P

With the help of the transformation σρ, we can now produce a relation between

the set of all partitions P and a subset P of P , defined thusly:

Definition B.19. P := α ∈ P |α does not have regular parts of the same q-length.

As an element of P we always imagine a partition α illustrated by the diagram

[α]∗q. It follows thus that:

P = α ∈ P |[α]∗q

does not have two residue-units in a row. B.19′

A diagram [α]∗q

with α ∈ P and ρα ∈ Nr

q−1has thus at least r rows. A row in

[α]∗q

which has no residue-units we will call singular.

As will similarly be shown, there corresponds to the set Pρk l

of B.10, in P , the set

Definition B.20. Pρk l

:= α ∈ P |ρα = ρ, [α]∗qhas k singular columns and

l singular rows.

87

Completely analogously to B.10 and the remarks on it, the sets Pρk l

, P r· l

, Pk l

and

so forth are also to be understood as unions of sets Pρk l

. Similarly to Pρ0 0

, the set Pρ0 0

is of one element, because

B.21. mρ :=((r − 1)q + ρ1, (r − 2)q + ρ2, . . . , ρr

)is the only element of Pρ

0 0. From B.21 it follows immediately:

Theorem 25.∣∣mρ∣∣ = (r2).

In analogy to B.12 we can thus write each partition α ∈ P in the form

B.22. α = αf

+ qαs

with certain unique partitions αf∈ P· 0 and qα

s∈ P 0(= P

0). The diagram[qαs

]q

thus

consists of the singular rows of [α]∗q. Obviously α is an element of Pρ

k l, exactly when α

f

is an element of Pρ· 0

and qαs

is an element of P 0

· l.

Remark: The decomposition of α ∈ P according to B.12 is not in all cases identical

with the decomposition of α in B.22. For example, for the partition α := (8, 1) one sees

immediately that for q = 3 one has αf

= (2, 1) and qαs

= (6). On the other hand one

has αf

= (5, 1) and qαs

= (3).

An immediate conclusion is the following Lemma:

Lemma 1. For all r ∈ N0, ρ ∈ Nrq−1

the transformation ερ : P 0

· ≤r→ P

ρ

· 0, defined by

ερ (qδ) = m

ρ⋃qδ is bijective, and it is true for all k ∈ N0, qδ ∈ P

0

· ≤rthat

(i)∣∣ερ (qδ)

∣∣q

=(r2)

+ |δ|

(ii) ερ(P

0

k≤r

)= P

ρ

k 0.

88

Because of Theorem 24, one will be able to write each flat partition α in the form

B.23. α = mρ 5 qβ

with ρ = ρα and a certain unique qβ ∈ P

0

·≤r. Thereby one has qβ =

(σρ)−1 (α).

Together with Lemma 1 it follows thence, that by means of the rule

B.24. φ′q(α) = φ

′q

(mρ 5 qβ

):= m

ρ⋃qβ

a bijective transformation φ′q

from P0

· 0to P· 0 is defined. φ′

qis thus the transformation

which, restricted to Pρ· 0

, completes diagram B.5 commutatively.

Fig. B.5.

Because of B.12 and B.22, one can lift φ′q

to a bijective transformation of P to

P , in that one, for arbitrary α ∈ P , sets

B.25. φq(α) := φ

′q

(αf

)+ qα

s.

With this we may now formulate in summation the following theorem:

89

Theorem 26. The transformation φq

: P → P , defined in B.25 and B.24 respectively,

is bijective, and it is true that for all k, l, r ∈ N0, ρ ∈ Nrq−1

, α ∈ Pρ:

(i)∣∣∣φq(α)∣∣∣q

=(r2)−∣∣mρ∣∣

q+ |α|

q

(ii) φq

(Pρ

k l

)= P

ρ

k l.

Proof. The number of parts of α is determined – due to the construction of φq

– in

agreement with the number of parts of φq(α). Since the number of regular parts is left

unaltered by application of φq

to α, α is an element of Pk · exactly when φ

q(α) lies in

Pk · . Everything else follows from Theorem 24 and Lemma 1.

B.4.4 A Generalization of the Conjugating Transformation C to P

Because of B.22 and Lemma 1 one can write any partition γ of P in an unam-

biguous way in the form

γ =(mρ⋃

qδ)

+ qγs

with ρ = ργ and qδ ∈ P 0

· ≤r. These allow us, with B.4, by means of the assignment

B.26. Cq(γ) =

(mρ + qδ

′)⋃qγ′s

( where qγ′s

:= q(γs

)′) to define a transformation C

qof P .

90

Example B.27. Set q ≥ 5 and

[γ]∗q

:=

q q q q q 4

q q q

q 2

q

1

.

Then one has[qγs

]q

=q q q

q

and [qδ]q

= q q q, and it follows that

[qγ′s

]q

=

q q

q

q

and

[qδ′]q

=

q

q

q

91

respectively. Thus one has[mρ + qδ

′]∗q

=

q q 4

q 2

q

q

q

1

and with this

[Cq(γ)]∗q

=

q q q q 4

q q 2

q q

q

q

1

.

It is itself easy, that the diagram[Cq(γ)]∗q

also arises from [γ]∗q

via

B.26′: S1 : mirror all units of [γ]∗q

about the main diagonal of [γ]∗q

and

S2 : Exchange the residue-units ρν

with ρr+1−ν for 1 ≤ ν ≤ r

2 .

Straightaway from B.26′ there results a picture of γ from Example B.27 under

Cq, as well, as follows:

92

[γ]∗q

=

q q q q q 4

q q q

q 2

q

1

−−−→S1

q q q q 1

q q 2

q q

q

q

4

−−−→S2

q q q q 4

q q 2

q q

q

q

1

=[Cq(γ)]∗q

(In many cases it is appropriate to use the definition B.26′).

Observation: The transformation rule given in B.26′ is not meaningful for diagrams [γ]q

with γ ∈ P \ P , as a diagram with two residue-units in one row would be converted via

application of S1 (and S2) into a diagram with two residue-units in one column.

Theorem 27. The transformation Cq

: P → P defined by B.26 (resp., B.26′) is

bijective, and it is true for all k, l, r ∈ N0, ρ ∈ Nrq−1

that:

(i) Cq

(Pρ

k l

)= P

ρ

l k

(ii) C2

q= id

P.

Proof. As it makes no difference in which order S1 and S2 are used, and since dual uses

of S1 (resp. S2) do not alter a diagram [γ]∗q, one has that C2

q= id

P. Part (i) of the

assertion follows immediately from the definition of Cq.

93

Obviously one has, for q = 1, P = P and C1 is identical to the conjugation

transformation C defined in Chapter 1.

B.4.5 The Bijection Lq

The transformation φq

defined in B.25 can be “shifted” by the bijection Cq

from

P to P :

Fig. B.6.

Thus we come now to the main result with this paragraph:

94

Theorem 28. The transformation Lq

:= φ−1

qCqφq

is a bijection of the set of all

partitions P , and one has for all k, l, r ∈ N0, ρ ∈ Nrq−1

, α ∈ P :

(i)∣∣∣Lq(α)∣∣∣q

= |α|q

(ii) Lq

(Pρ

k l

)= P

ρ

l k

(iii) L2

q= id

P.

Proof. (i) follows from Theorem 26 (i), because for γ in P one has∣∣∣Cq(γ)∣∣∣q

= |γ|q.

(ii) holds from Theorem 26 (ii) and Theorem 27 (i).

(iii) follows from Theorem 27 (ii).

In the following we often consider the restriction of Lq

to P (n). We write then

Lq,n

for Lq|P (n) for short. Note that, due to Theorem 28 (i) and (ii), L

q(P (n)) = P (n).

Finally some special cases are to be mentioned:

B.28. Set ρ =(ρ1, . . . , ρr

)and ρ1 < ρ2 < · · · < ρ

r. Then it holds that P

ρ = Pρ

and φq|ρP

= idPρ. For all α ∈ Pρ it is true then that L

q(α) = C

q(α).

B.29. For q = 1, ρ = ∅ is the unique residue-vector and thus by B.28 L1 = C1 .

Since C1 is identified with the conjugating transformation C in Chapter 1, we thus

have L1 (α) = α′ for all α ∈ P .

B.30. For q ≥ n the sets Pρ(n) are single-element for all r ∈ N0, ρ ∈ Nr

q−1,

because for ρ 6= (n) , we simply have Pρ(n) = mρ . Because of Theorem 28 (ii) we

thus have Lq,n

= idPρ(n).

95

We explain now in summary how the partition Lq(α) is made from α:

A1 One determines ρ = ρα and constructs the diagram

[mρ]∗q.

A2 One removes from [α]q

all strips and builds from these the diagram[qαs

]q

(see B.12).

Then there remains a flat diagram[αf

]q.

A3 a) One determines those singular columns of[αf

]q

which, when one removes them

from[αf

]q, leave a flat diagram, and combines these columns with

[mρ]∗q.

b) Subsequently one removes from left to right sequentially all angles, writes the q-

units of each angle as a column by itself and combines these with the diagram that was

received from a). In this way the diagram[mρ⋃

qβ]∗q

=[φq

(αf

)]∗q

develops.

A4 The rows of[qαs

]q

are inserted between the rows of[mρ⋃

qβ]∗q

in such a way that a

diagram results. If one has not made an error, only the residue diagram[mρ]q

remains

of [α]q, while one gets the diagram

[φq(α)]∗q

from[mρ]∗q.

B1 The diagram[φq(α)]∗q

is reflected about the main diagonal.

B2 One permutes the residue units ρ1, . . . , ρr so that the original sequence is restored.

In this way the diagram[Cqφq(α)]∗q

=: [γ]∗q

develops.

C1 One removes the singular rows from [γ]∗q

and builds from them the diagram[qγs

]q.

Then the diagram[γf

]∗q

remains, consisting only of the regular rows of [γ]∗q.

C2 The single columns of[γf

]∗q

are removed sequentially from left to right and inserted

as – depending on q-length – angles or columns into [m]q. Thus we get the diagram[

mρ 5 qδ

]q

=[φ−1

q

(γf

)]q

.

C3 The rows of[qγs

]q

are inserted as strips into[mρ 5 qδ

]q. Thus results the diagram[

φ−1

q(γ)]q

.

96

C4 The diagram[φ−1

q(γ)]q

=[φ−1

qCqφq(α)]q

=[Lq(α)]q

gives us the desired parti-

tion.

97

Fig. B.7. A sketch of the process.

98

Example B.31. Set q ≥ 5. For q = 5 in the example we have

α =(42, 39, 30, 25, 23, 20, 16, 10, 7, 52)

and

Lq(α) = (50, 37, 29, 28, 25, 16, 15, 12, 10).

Fig. B.8.

99

Fig. B.9.

Fig. B.10.

100

Fig. B.11.

B.5 Some Counting Theorems for Partitions

In the modular representation theory of symmetric groups there arises, due to

a central result, a one-to-one mapping between the q-flat partitions and the q-modular

irreducible representations for prime numbers q. Since the number of q-modular irre-

ducible representations also agrees with the number of q-regular conjugation clases, one

gets from this representation theory connection, as a counting theorem, a partial result

of an enumeration by Euler (for q prime) which says that for all q, n ∈ N the set of q-flat

partitions of n is the same size as the set of q-regular partitions of n.

By evaluation of the transformation Lq

this result of Euler’s will now be extended

and refined. As special cases there will also result from this identities which were found

by Sylvester (Chapter 4) and Fine.

From Theorem 28 there follows immediately by suitable summation:

101

Theorem 29. For all k, l, r ∈ N0, ρ ∈ Nrq−1

one has:

(i)∣∣∣∣Pρk l

(n)∣∣∣∣ = ∣∣∣∣Pρ

l k(n)∣∣∣∣

(ii)∣∣∣∣P rk l

(n)∣∣∣∣ = ∣∣∣∣P r

l k(n)∣∣∣∣

(iii)∣∣∣Pk l

(n)∣∣∣ = ∣∣∣P

l k(n)∣∣∣

or in words:

(iii)′ The number of all q-diagrams of n with k singular columns and l edge

units is equal to the number of all q-diagrams of n with l singular columns and k edge

units.

The result Theorem 29 (iii)′ shows perhaps most clearly that the transformations

Lq

for q > 1 can be interpreted as generalizations of the conjugating transformation C,

since Lq

causes a permutation of q-singular columns and/or angles and q-strips in the

diagram [α]q. With increasing q and fixed n the transformations L

q,nwill become more

and more similar to the identity, and finally coincide for q ≥ n.

From Theorem 29 follows by further specialization:

Corollary 7. For all k, n ∈ N0 it holds that:

(i)∣∣∣Pk 0(n)

∣∣∣ = ∣∣∣P0 k(n)∣∣∣

(ii)∣∣∣Pk ·(n)

∣∣∣ = ∣∣∣P· k(n)∣∣∣

(iii) (Euler, Glaisher) The number of all q-regular partitions of n is equal to the

number of all q-flat partitions of n.

102

Proof. (i) One sets l = 0 in Theorem 29 (iii).

(ii) For each k one sums in Theorem 29 (iii) over l.

(iii) follows from Corollary 7 (ii) for k = 0.

With Definition B.11 and Theorem 19 we can formulate Corollary 7 as follows:

Corollary 7′. (i)′ The number of all partitions of n with α′q−α′

q+1+α′

2q−α′

2q+1±· · · = k

and∣∣∣α1 − α2

∣∣∣q

+∣∣∣α2 − α3

∣∣∣q

+ · · · = 0 is the same as the number of partitions of n with

α′q− α′

q+1+ α′2q− α′

2q+1± · · · = 0 and

∣∣∣α1 − α2

∣∣∣q

+∣∣∣α2 − α3

∣∣∣q

+ · · · = k.

(ii)′ The number of all partitions of n with αq− α

q+1 + α2q − α2q+1 ± · · · = k

is equal to the number of all partitions of n with∣∣∣α1 − α2

∣∣∣q

+∣∣∣α2 − α3

∣∣∣q

+ · · · = k.

In the following, set pk l

(n) to be the number of all partitions of n with k parts

and with α1 = l. Obviously pk l

(n) is also the number of all singular q-diagrams of n

q-units with k columns and l lines:

B.32. pk l

(n) =∣∣∣∣P 0

k l(qn)

∣∣∣∣.Accordingly we set also then p

k≤l(n) :=∣∣∣∣P 0

k≤l(qn)

∣∣∣∣ (=∑l

j=0pk j

(n))

,

pk ·(n) :=

∣∣∣∣P 0

k ·(qn)

∣∣∣∣, etc(see the remarks to Definition B.11). For example, one has

p≤k ·(n) the number of all partitions of n with at most k parts.

We have seen based on B.12 and Theorem 24 that each partition α ∈ Pρk l, ρ =(

ρ1, . . . , ρr

)can be written uniquely in the form

B.33. α =(mρ 5 qβ

)+ qγ

103

with qβ ∈ P 0

k≤rand qγ ∈ P 0

· l. Since also to each pair of partitions (qβ, qγ) ∈ P 0

k≤r×P 0

· l

belongs in the reverse way a partition α ∈ Pρk l

, we get with B.32:

Theorem 30. Set ρ =(ρ1, . . . , ρr

)and n =

∣∣mρ∣∣+ qf with f ∈ N0. Then it holds that

for all k, l ∈ N0:

∣∣∣∣pρk l

(n)∣∣∣∣ = ∑

g,h∈N0g+h=f

pk≤r(g) p· l(h).

We present now a connection between the q-length∣∣∣α1

∣∣∣q

of the largest part of α

and the width of Lq(α). Set additionally for α:

B.34. gα

:=∣∣∣i|1 ≤ i ≤ w, 0 < r

q

(αi

)≤ r

q

(αi−1

)∣∣∣

(setting α0 := q∣∣∣α1

∣∣∣q

+ (q − 1)). gα

gives thus the number of all regular parts(αf

)i

of αf

for which∣∣∣∣(αf)i

∣∣∣∣q

=∣∣∣∣(αf)i−1

∣∣∣∣q

holds (with 1 ≤ i ≤ w). As one easily sees, it

holds for α ∈ P r with r ∈ N0 that∣∣∣∣(αf)1

∣∣∣∣q

+ gα

= r and thus because of B.12, for

α ∈ P r· l

(r, l ∈ N0),

B.35.∣∣∣α1

∣∣∣q

+ gα

= r + l.

Thus results for all v ∈ N0:

∣∣∣α1

∣∣∣q

+ gα

= v ⇔ the first column of[φq(α)]qcontains v units

⇔ Cqφq(α) has v parts

⇔ Lq(α) has v parts.

104

Since Lq

transforms at the same time the flat partitions of P (n) to the regular

partitions of P (n), we have proved the following theorem:

Theorem 31. (i) For all q, v ∈ N0 it holds that the number of all partitions on n with

v parts is equal to the number of all partitions of n with∣∣∣α1

∣∣∣q

+ gα

= v.

(ii) For all q, v ∈ N0 it holds that the number of all flat partitions of n with v

parts is equal to the number of all regular partitions of n with∣∣∣α1

∣∣∣q

+ gα

= v.

Remarks: (i) From part (i) of Theorem 31 one infers also that the number of all partitions

of n with∣∣∣α1

∣∣∣q

+ gα

= v is equally large for all q ∈ N and thus independent of q.

(ii) For q = 1 it holds for all α ∈ P (n) that qα

= 0 and∣∣∣α1

∣∣∣1

= α1. From

Theorem 31 (i) then follows the well-known fact that the number of all partitions of n

with largest part v is the same as the number of all partitions of n with v parts.

Part (ii) of Theorem 31 should be formulated separately for q = 2. If α =(α1, . . . , αw

)is a 2-regular partitions, it is obvious that

∣∣∣α1

∣∣∣2

= α1−12 and g

α= w.

Thus follows for all v ∈ N0:

∣∣∣α1

∣∣∣2

+ gα

= v ⇔ α1 = 2v − 2w + 1 .

If one notes now the fact that a partitions α is 2-flat and consists of v parts exactly

when α′ is a partition of distinct parts with largest part v, then from this results:

Corollary 8. The number of all partitions of n with distinct parts and with largest part

v is equal to the number of all partitions of n with odd parts, whose largest part is equal

to 2v − 2w + 1, if w is the number of all parts.

105

B.6 The Special Case q = 2

The case q = 2 deserves special attention, since one can quickly grasp the 2-

modular residue diagram: the 2-residue-vectors are all of the form ρ = (1r) with r ∈ N

and therefore are always identifcal to their residue diagram: mρ = ρ = (1r).

For the derivation of the following counting theorems it is advisable to introduce

a furtherm term: if α = (α1, . . . , αw) is a partitions, then one divides the set of parts

of α into maximal blocks of successive parts of the same parity. Starting from the rear

(the smallest part) the regular blocks (blocks formed from regular parts) are B1, B3, . . . ,

and the singular blocks are B0, B2, B4, . . . if 2 | αw

, or B2, B4, . . . if 2 - αw

. We call

bi:=∣∣∣Bi

∣∣∣ the width of Bi.

Example B.36. The following figures demonstrate two possibilities for blocks.

Fig. B.12.

106

Fig. B.13.

Lemma 2. In 2-flat partitions two parts αi

and αj

belong to the same block only if

αi= α

j. For the parts α

jof a partition α ∈ P· 0 it thus holds:

αj∈ B2i+1 ⇐⇒

∣∣∣αj

∣∣∣2

= i anf 2 - αj,

αj∈ B2i ⇐⇒

∣∣∣αj

∣∣∣2

= i and 2 | αj.

Set p to be the number of singular blocks of a 2-flat partitions α. Then by Lemma

2 α is clearly specified by the vector

B.37. bα

:=(b1, b2, . . . , b2p+ε

)wherein 2p+ ε, ε ∈ 0, 1, is the number of blocks of α.

In light of this, the partition φ2(α) is thus specified by the vector

B.38. aα

:=(α1, α2, . . . , αr

)by Lemma 1, wherein a1 > · · · > a

rare the indices of the regular parts of φ2(α) (see

Example B.39).

107

Lemma 3. Set α ∈ P r· 0

, aα

and/or bα

as in B.37 and/or B.38, and a1 < a2 < . . . the

sequence of indices of singular parts of φ2(α). Then follows:

(i)∣∣∣a

ν|aν≤ r

∣∣∣ = p .

(ii) For all 1 ≤ ν ≤ p it holds (setting a0 := 0) that

b2ν−1 = aν− a

ν−1 and b2p+ε = r − ap, in the case 2 - α1 .

(iii) For all 1 ≤ ν ≤ p− 1 it holds that

b2ν = aν− a

ν+1and b2p = ap− r.

Proof. Since φ2(α) lies by supposition in P r· 0

, it holds by Lemma 1 that φ2(α) = m(1r)∪

qβ with qβ ∈ P0

· ≤r. Therefore β

i=∣∣∣∣φ2(α)

ai

∣∣∣∣2

is the number of 2-units in the ai-th

column of[φ2(α)

]∗2. For the singular parts of φ2(α) with ν ≥ 1 thus arises

βν

= r − aν

+ ν (1)

and thus (setting β0 := r, a0 := 0 ):

βν−1 − βν = a

ν− a

ν−1 − 1 . (2)

Because of (1) we have for all ν ≥ 1:

r − aν≥ 0 ⇐⇒ β

ν− ν ≥ 0 .

Thus part (i) follows from the statement of Corollary 6.

108

As the construction of φ−1

2and/or σ(1r) shows, the block B2ν−1 , which consists

of βν− β

ν−1 + 1 parts, develops – if βν− ν ≥ 0 – when inserting the ν-th angle in[

β(1r),ν−1

], to the right of this angle. Thus it follows from (2) that b2ν−1 = a

ν−a

ν−1.

The remaining statements directly follow.

Example B.39. Set the following:

Fig. B.14.

Then r = 7 and bα

= (2, 3, 1, 1, 1, 2, 3, 2). By Lemma 3, aα

= (15, 12, 11, 9, 6, 5, 1)

and thus we have the following.

109

Fig. B.15.

The following theorem describes how aα

and bα

depend on each other.

Theorem 32. If one sets, for 1 ≤ i ≤ p , si

:=i∑

ν=1b2ν−1 and s

′i

:=i∑

ν=1b2p−2ν+2 ,

then the elements of aα

result from the elements of bα

as follows:

a1, . . . , ar = (1, . . . , r \ s1, . . . , sp) ∪ r + s′1, . . . , r + s

′p .

Proof. The statement follows directly from Lemma 3 by induction.

We get a further lemma from Theorem 26: according to B.38, aigives the number

of units in the i-th regular line of[φ2(α)

]∗2. On the other hand set now a

′i(for 1 ≤ i ≤ r

) to be the number of units in the i-th regular column of[φ2(α)

]∗2. The numbers a

iand

a′i

we then arrange in the matrix pattern

B.40. T(φ2(α)

):=

a1 . . . ar

a′1

. . . a′r

together. For example, we have for α from Example B.39

110

T(φ2(α)

):=

15 12 11 9 6 5 1

7 6 5 4 3 2 1

.

From Theorem 26 follows now immediately, since C2 conjugates the 2-diagram,

that this means an exchange of the ai

with the a′i:

Lemma 4. Set r ∈ N0 , a1 > . . . ar

and a′1< . . . a

′r

with ai, a′i∈ N . Then it holds

that the number of all partitions of n with T(φ2(α)

):=

a1 . . . ar

a′1

. . . a′r

is equal to the

number of all partitions of n with T(φ2(α)

):=

a′1

. . . a′r

a1 . . . ar

.

Lemma 5. If α = (α1, . . . , αr) is a 2-regular partition, then one has

T(φ2(α)

)=

r r − 1 . . . 1∣∣∣α1

∣∣∣2

+ r∣∣∣α2

∣∣∣2

+ r − 1 . . .∣∣∣αr

∣∣∣2

+ 1

.

Proof. Since αf

= (1r) = m(1r) and α

s=(∣∣∣α1

∣∣∣2, . . . ,

∣∣∣αr

∣∣∣2

)it follows that

φ2(α) = m(1r) + 2α

s

= (2r − 1, . . . , 1) + (2∣∣∣α1

∣∣∣2, . . . , 2

∣∣∣αr

∣∣∣2)

= (2∣∣∣α1

∣∣∣2

+ 2r − 1, . . . , 2∣∣∣αr

∣∣∣2

+ 1) .

111

Since 2-flat and 2-regular partitions α are clearly fixed by the matrix T(φ2(α)

),

one can by use of Theorem 32 give explicitly the image of α ∈ P· 0 under L2, because we

have from Lemma 4:

B.41. T(φ2(α)

)=

a1 . . . ar

r ... 1

⇐⇒ T(C2

(φ2(α)

))=

r . . . 1

a1 . . . ar

⇐⇒ L2(α) = (2a1 − (2r − 1), . . . , 2ar− 1) .

With the assistance of Theorem 32 (and/or Lemma 3) and B.41 we can easily

state some counting theorems:

Theorem 33. The number of all partitions of n of distinct odd parts is equal to the

number of all 2-flat partitions of n with the following characteristics:

(i) All parts of odd length except α1 occur at most twice.

(ii) All parts of even length except α1 occur at least twice.

(iii) α1 occurs at most once, if α1 is odd.

Proof. Set α ∈ P· 0(n) , γ := L2(α)(∈ P0 ·

), and set b

α= (b1, . . . , b2p+ε) and a

α=

(a1, . . . , ar) . The theorem results from the following chain of equivalences:

( b2j−1 ≤ 2 for all 1 ≤ j ≤ p ) and ( b2j ≥ 2 for all 1 ≤ j ≤ p − 1 + ε ) and (

b2p+1 ≤ 1(withb2p+1 := 0, incaseε = 0) )

⇐⇒ ai≥ a

i+1 + 2 for all 1 ≤ i ≤ r (calling ar+1 := −1) by Theorem 32

⇐⇒ 2ai− (2r + 1− 2i) ≥ 2a

i+1 − (2r + 1− 2(i+ 1)) + 2 for all 1 ≤ i ≤ r

112

⇐⇒ γi≥ γ

i+1 + 2 for all 1 ≤ i ≤ r (calling γr+1 := 0) by B.41

⇐⇒ γi6= γ

i+1 for all 1 ≤ i ≤ r, since 2 - γi.

The following theorem can be proven completely analogously:

Theorem 34. The number of all 2-regular partitions of n with αi− α

i+1 ≤ 2 for all

a ≤ i ≤ w (setting αw+1 := 0) is equal to the number of 2-flat partitions of n with the

following characteristics:

(i) All parts of even length except α1 occur at most twice.

(ii) All parts of odd length except α1 occur at least twice.

(iii) α1 occurs at most once, if α1 is even.

The following result comes from J. J. Sylvester [14]:

Theorem 35. Let Ak(n) be the set of all 2-flat partitions of n of exactly k − 1 blocks

Bibesides B2p+ε with b

i> 1, and let C

k(n) be the set of all partitions of n of exactly k

different odd parts. Then∣∣∣Ak(n)∣∣∣ = ∣∣∣C

k(n)∣∣∣.

Proof. Set α ∈ Ak(n) with b

αand a

αlike above. We still set a

r+1 := −1. Then we

have:

113

k =∣∣∣j|1 ≤ j ≤ 2p+ ε, 2 - j, b

j> 1

∣∣∣+ ε

+∣∣∣j|1 ≤ j ≤ 2p+ ε, 2 | j, b

j> 1

∣∣∣+ 1− ε

=∣∣∣j|a

j≤ r, a

j> a

j+1 + 1, 1 ≤ j ≤ r∣∣∣

+∣∣∣j||a

j> r, a

j> a

j+1 + 1, 1 ≤ j ≤ r∣∣∣ by Theorem 32

=∣∣∣j|a

j> a

j+1 + 1, 1 ≤ j ≤ r∣∣∣

=∣∣∣j|2a

j− 2r + 2j − 1 > 2a

j+1 − 2r + 2(j + 1)− 1, 1 ≤ j ≤ r∣∣∣ .

From B.41 it follows now, that a partition is an element of Ak(n) exactly when

L2(α) consists of exactly k different parts of odd length.

In the counting theorems 33 through 35 we in each case compared subsets of P0 ·.

In the following theorem this is no longer the case:

Theorem 36. Let R(n) be the set of all partititions of n with 2 | α1 and without

consecutive parts of odd length, and let S(n) be the set of all partitions of n whose

smallest part of odd length is at least twice as large as the number of all parts of odd

length. Then |R(n)| = |S(n)|.

Proof. With B.40 there results:

114

|R(n)| =∣∣∣α ∈ P (n)|b

α= (b0, b2p), b2i−1 ≤ 1 for 1 ≤ i ≤ p

∣∣∣=

∣∣∣∣∣∣∣∣α ∈ P (n)||T (φ2(α)) =

a1 . . . ar

a′1

. . . a′r

, r ∈ N0, ar > r

∣∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣∣α ∈ P (n)||T (φ2(α)) =

a′1

. . . a′r

a1 . . . ar

, r ∈ N0, a′r> r

∣∣∣∣∣∣∣∣= |S(n)| .

Example B.42. n = 9:

R(n) = (8, 1), (6, 3), (6, 2, 1), (42, 1), (4, 3, 2), (4, 22

, 1)(24, 1)

S(n) = (9), (7, 2), (6, 3), (5, 4), (5, 22), (4, 3, 2), (3, 23)

B.7 The Fixed Points of Lq

If one writes the transformation Lq,n

= Lq|P (n) , understood as a permutation

of the full set P (n), as a product of disjoint cycles, then only cycles of length 1 or 2 arise

since L2

q,n= id

P (n). For the determination of the cycle-structure of Lq,n

it is sufficient

to know the number of fixed points of Lq,n

.

Since for each r ∈ N0, ρ ∈ Nrq−1

the set Pρ(n) is invariant under Lq,n

, it is

sufficient to regard the restriction of Lq,n

to Pρ(n). Because Lq

= φ−1

qCqφq, α ∈ Pρ is

115

a fixed point of Lq

exactly when φq(α) (∈ Pρ) is fixed by C

q. This latter is exactly the

case when

B.43. (i) the q-units in[φq(α)]∗q

are symmetrically arranged about the main diagonal,

and

(ii) the places of the residue-units in[φq(α)]∗q

are arranged in mirror-image to the

main diagonal. (It is however not the case that one needs ρi= ρ

r+1−i for 1 ≤ i ≤ r .)

Following the case of q = 1, we will want to call q-diagrams with the characteristics

B.43 (i) and (ii) (and/or the associated partitions in P ), q-self-conjugate.

Example B.44. For example, the following q-diagram is q-self-conjugate, when q ≥ 5:

Each diagram [γ]∗q

with γ ∈ P can – as in Example B.44, on the basis of the

nested lines – be understood as a combination of “hooks” nested within each other,

whose corners lie on the main diagonal. The condition B.43 is then obviously equivalent

to:

B.43′ (i) Both legs of each of the hooks in [γ]∗q

contain the same number of q-units.

116

(ii) Each hook contains either zero or exactly two residue units, with one excep-

tion: if 2 - r, then the smallest hook consists of exactly one residue unit.

If a hook of a q-self-conjugate partition contains residue units, then it has at least

four, otherwise at least two units more, than the next smaller hook.

For s ∈ N0 let Aρs

be the set of all q-self-conjugate partitions γ ∈ Pρ, whose

q-diagram [γ]∗q

consists of s hooks. In counting the elements of Aρs(n), we limit ourselves

at first to the case wherein r is even: let r = 2r′, with r′ ∈ N0. If γ ∈ Aρs, then

[(qs)s

]∗q

is the largest square consisting of q-units which is contained in [γ]∗q. Obviously [γ]∗

q(with

given fixed ρ and s) is already clearly fixed by the diagram [γ]∗q

underneath of[(qs)s

]∗q.

We have that γ is an element of Pρ′

≤s−r′ ·, wherein one should let ρ′ := (ρ1, . . . , ρr′).

Vice versa, to each partition δ ∈ Pρ′

≤s−r′ ·belongs an element γ of Aρ

swith γ = δ; the

element of Aρs

is thus characterized by the partition from Pρ′

≤s−r′ ·. From

∣∣∣mρ∣∣∣q

= 2∣∣∣∣mρ′∣∣∣∣

q+ r′2

follows for the q-weight of α := φ−1

q(γ) ∈ Pρ:

|α|q−∣∣∣mρ∣∣∣

q= |γ|

q−∣∣∣mρ∣∣∣

q

= 2|γ|q

+ s2 −

∣∣∣mρ∣∣∣q

= 2

(|γ|q−∣∣∣∣mρ′∣∣∣∣

q

)+ s

2 − r′2 .

117

Together with Theorem 30 results now part (i) of the following theorem:

Theorem 37. Let ρ = (ρ1, . . . , ρr) and n =∣∣mρ∣∣ + qf . Then one has for the number

of fixed points of Lq|Pρ(n):

(i)Fix(Lq|Pρ(n)) =

∑s≥r′

∑g,h∈N0

g+h=f+r′2−s

2

2

P≤r′ ·(g)P≤s−r′ ·(h)

in the case r = 2r′

(ii)Fix(Lq|Pρ(n)) =

∑s≥r′

∑g,h∈N0

g+h=f+r′2−s

2

2

P≤r′−1 ·(g)P≤s−r′ ·(h)

in the case r = 2r′ − 1 .

The only difference between the proof of (ii) and that of (i) lies in the fact that

for odd r the largest sequence in[φq(α)]∗q

contains a residue unit in the lower right hand

corner.

B.8 Groups of Permutations of P (n)

We have defined the transformation Lq,n

(= Lq|P (n)) in chapter 2 and now want

to ascertain the permutation group generated by Li,n, . . . , L

n−1,n for 1 ≤ i ≤ n− 1. In

addition we define for i ∈ N (and for previously fixed n ∈ N ):

B.45. Ωi:= α ∈ P (n)|α1 ≥ i

and

118

B.46. Si := π ∈ SP (n)|π(α) = α for all α ∈ P (n) \ Ω

i (' SΩi

).

We can now formulate the following theorem.

Theorem 38. (i) For n ≥ 2 and 1 ≤ i ≤ n− 1 it holds that

〈Li,n, . . . , L

n−1,n〉 = Si, in the case i 6= n− 2 .

(ii) For n ≥ 3 it holds that:

〈Ln−2,n, Ln−1,n〉 = DΩn−2

.

Here DΩn−2is the dihedral group of the four points of Ω

n−2, when n ≥ 4. For n = 3,

DΩn−2= S

3 ' S3 .

For the proof of this theorem we need a lemma that guarantees the existence of

certain fixed points of Lq,n

:

Lemma 6. For n ≥ 6 and 1 ≤ q ≤ n − 3, Lq,n

has at least one fixed point α0 with

α0 ∈ Ωq+1.

Proof. There are three cases to separate:

(i) 3 ≤ q ≤ n− 3: the partition α0 := (q + 1, 2, 1n−q−3) is q-flat and q-regular

and is thus a fixed point from Ωq+1.

(ii) q=2: One checks easily that α0 = (4, 1n−4) will be fixed by L2.

(iii) q=1: For n = 2r+ 1 the partition α0 := (r+ 1, 1r) is self-conjugate and thus

a fixed point of L1,n. For n = 2r, α0 := (r, 2, 1r−2) will be fixed by L1,n.

119

Proof of Theorem 38. Let 1 ≤ q ≤ n − 1 and α = (α1, . . . , αw) ∈ P (n). In some cases

the image of α under Lq

can be specified immediately:

B.47. For α /∈ Ωq, α = m

α and so Lq,n

(α) = α.

B.48. If α ∈ Ωq\ Ω

q+1, then α = (qk, αk+1, . . . , αw) with k ≥ 1 and α

k+1 < q , if

w > k. Because α ∈ Pw−kk ·

we have that Lq,n

(α) ∈ Pw−k· k

and so

Lq,n

(α) = (kq + αk+1, αk+2, . . . , αw)

(setting αk+1 := 0, in the case w = k.)

We have Ωn−1 = (n), (n− 1, 1) and so by B.47 and B.48:

Ln−1,n = ((n− 1, 1)(n)) .

Part (i) of the statement thus holds for i = n− 1.

For n ≥ 4 one has Ωn−2 = (n), (n− 1, 1), (n− 2, 12), (n− 2, 2) and thus again

by B.47 and B.48

Ln−2,n = ((n− 2, 12), (n− 1, 1))((n− 2, 2)(n)) .

Since two involutions π1 and π2 generate a dihedral group of order 2m, where

m is the order of π1π2, there follows here from the fact that Ln−2,nLn−1,n = ((n −

1, 1)(n− 2, 2)(n)(n− 2, 12)):

120

〈Ln−2,n, Ln−1,n〉 = 〈L

n−2,nLn−1,n, Ln−1,n〉 = DΩn−2.

Thus is part (ii) of the statement proven, because for n = 3 one has L1,n =

((13)(3)) and L2,n = ((2, 1)(3)).

The proof of Theorem 38 for 1 ≤ i ≤ n − 3 and n ≥ 6 now follows by induction

on i:

The statement is true for i = n − 3: we have namely Ωn−3 = Ω

n−2 ∪ (n −

3, 13), (n− 3, 2, 1), (n− 3, 3) and so Ln−3,n = ((n− 3, 13)(n− 2, 12))((n− 3, 2, 1)(n−

1, 1))((n−3, 3)(n)) by B.47 and B.48. Since Ln−3,n ·Ln−2,n ·Ln−1,n is a cycle of length

7 ( =∣∣∣Ωn−3

∣∣∣ ), this together with Ln−1,n generates the symmetric group S3(' S7).

Assume the statement holds for i+1 ( i+1 ≤ n−3 ). By Lemma 6, one gets then

an α0 ∈ Ωi+1 with L

i,n(α0) = α0. Since for fixed ω0 ∈ Ω(|Ω| ≥ 2) the transpositions

(ω, ω0) with ω ∈ Ω generate the symmetric group SΩ, it is sufficient to show that for all

α ∈ Ωi

the transposition (α, α0) lies in 〈Li,n, . . . , L

n−1,n〉. For α ∈ Ωi+1 this is already

fulfilled by the induction hypothesis. For α ∈ Ωi\Ω

i+1 meanwhile one has by B.48 that

Li,n

(α) ∈ Ωi+1 and it follows that

(α, α0) = (L2

i,n(α), L

i,n(α0))

= Li,n

(Li,n

(α), α0)L−1

i,n∈ 〈L

i,n, . . . , L

n−1,n〉 .

121

As the following list proves, the statement also holds for the remaining cases

n = 4; i = 1, n = 5; i = 1, and n = 5; i = 2.

Corollary 9. For n ≥ 2 we have:

〈L1,n, . . . , Ln−1,n〉 = SP (n) .

Remark: Since the proof of Theorem 38 is constructive, one can for each bijective trans-

formation f : A→ B with A,B ⊆ P (n) produce a sequence of natural numbers q1, . . . , qr

with 1 ≤ qi≤ n− 1 and r ∈ N such that

B.49. f = Lq1|A· · · · · L

qr|A

holds.

The following list contains, for 2 ≤ n ≤ 9, the transformations Lq,n

for all relevant

q ∈ N. (The fixed points of Lq,n

are not noted. Since no misunderstanding can develop

here4, for the sake of clarity, the commas and brackets of partitions were omitted.)

n = 2

P (2) = 12, 2

L1,2 = (12, 2)

4Translator’s note: Since all the parts are single digits.

122

n = 3

P (3) = 13, 21, 3

L1,3 = (13, 3)

L2,3 = (21, 3)

n = 4

P (4) = 14, 212

, 22, 31, 4

L1,4 = (14, 4)(212

, 31)

L2,4 = (212, 31)(22

, 4)

L3,4 = (31, 4)

123

n = 5

P (5) = 15, 213

, 221, 312, 32, 41, 5

L1,5 = (15, 5)(213

, 41)(221, 32)

L2,5 = (213, 312)(221, 5)(32, 41)

L3,5 = (312, 41)(32, 5)

L4,5 = (41, 5)

n = 6

P (6) = 16, 214

, 2212, 23

, 313, 321, 33

, 412, 42, 51, 6

L1,6 = (16, 6)(214

, 51)(2212, 42)(23

, 32)(313, 412)

L2,6 = (214, 313)(2212

, 51)(23, 6)(321, 32)

L3,6 = (313, 412)(321, 51)(32

, 6)

L4,6 = (412, 51)(42, 6)

L5,6 = (51, 6)

124

n = 7

P (7) = 17, 215

, 2213, 231, 314

, 3212, 322

, 321, 413, 421, 43, 512

, 52, 61, 7

L1,7 = (17, 7)(215

, 61)(2213, 52)(231, 43)(314

, 512)(3212, 421)(322

, 321)

L2,7 = (215, 314)(2213

, 512)(231, 7)(3212, 321)(322

, 61)(421, 52)

L3,7 = (314, 413)(3212

, 512)(322, 52)(321, 7)(43, 61)

L4,7 = (413, 512)(421, 61)(43, 7)

L5,7 = (512, 61)(52, 7)

L6,7 = (61, 7)

125

n = 8

P (8) = 18, 216

, 2214, 2312

, 24, 315

, 3213, 3221, 3212

, 322,

414, 4212

, 422, 431, 42

, 513, 521, 53, 612

, 62, 71, 8

L1,8 = (18, 8)(216

, 71)(2214, 62)(23

, 12, 53)(24

, 42)(315, 612)

(3213, 521)(3221, 431)(3212

, 422)(414, 513)

L2,8 = (216, 315)(2214

, 513)(2312, 71)(24

, 8)(3213, 3212)

(3221, 53)(322, 521)(4212, 612)(422

, 62)

L3,8 = (315, 414)(3213

, 513)(3221, 521)(3212, 71)(322, 8)(431, 42)(53, 62)

L4,8 = (414, 513)(4212

, 612)(422, 62)(431, 71)(42

, 8)

L5,8 = (513, 612)(521, 71)(53, 8)

L6,8 = (612, 71)(62, 8)

L7,8 = (71, 8)

126

n = 9

P (9) = 19, 217

, 2215, 2313

, 241, 316, 32214

, 32213, 3221, 33

, 415, 4213

,

4221, 4312, 432, 421, 514

, 5212, 522

, 531, 54, 613, 621, 63, 712

, 72, 81, 9

L1,9 = (19, 9)(217

, 81)(2215, 72)(2313

, 63)(241, 54)(316, 712)

(3214, 621)(32212

, 531)(323, 421)(3213

, 522)(3221, 432)(415, 613)

(4213, 5212)(4221, 4312)

L2,9 = (217, 316)(2215

, 72)(2313, 712)(241, 9)(3214

, 3213)

(32212, 531)(323

, 81)(3221, 33)(4213, 613)(4221, 72)

(4312, 5212)(432, 63)(421, 54)(522

, 621)

L3,9 = (316, 415)(3214

, 514)(32212, 5212)(323

, 522)(3213, 712)

(3221, 81)(33, 9)(4312

, 421)(432, 72)(531, 54)

L4,9 = (415, 514)(4213

, 613)(4221, 621)(4312, 712)(432, 72)

(421, 9)(54, 81)

L5,9 = (514, 613)(5212

, 712)(522, 72)(531, 81)(54, 9)

L6,9 = (613, 712)(621, 81)(63, 9)

L7,9 = (712, 81)(72, 9)

L8,9 = (81, 9)

127

Stockhofe’s Bibliography

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plications, Addison-Wesley Publishing Company, London (1976).

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of Mathematics, Oxford Series (2) 17 (1966), 335–338.

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616–618.

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lished.

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(1959), 86–94.

• James, G.D. & Kerber, A., Representation Theory of the Symmetric Groups, En-

cyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Com-

pany, London (1981)

128

• MacMahon, P.A., Combinatory Analysis I,II, Cambridge University Press, London

and New York (1915/1916)

• MacMahon, P.A., The Theory of Modular Partitions, Proc. Cam. Pil. Soc. 21

(1923), 197–204.

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330.

129

About the Author

May 5 1952 Born in Haltern, district of Recklinghausen

Father: Josef Stockhofe, farmer

Mother: Cacilia Stockhofe, nee Paßmann

Religion: Roman Catholic

Schools Attended:

1958–1962 kath. Volkschule Lavesum

1962–1970 Neusprachliches Gymnasium Haltern

June 1970 University entrance exams

Course of Study:

WS 1970/71 Study of the field of civil engineering at

the RWTH Aachen

Apr. 1 1971–Sep. 30 1972 Service with the German Federal Armed Forces

WS 1972/73–SS 1978 Mathematics and physics study at the RWTH Aachen

July 1978 Main diploma examination in mathematics

November 1978 first philological state examination in math/physics

Professional Activity:

Sep. 16 1978–Spe. 30 1979 Trustee of the post of scientific assistant to

Chair D for Mathematics at the RWTH Aachen

since Oct. 1 1979 assistant lecturer at the Chair D for mathematics

at the RWTH Aachen

130

(Translator’s Note: I may have failed to issue a precise translation of the names

of several of Mr. Stockhofe’s posts, but hope that the information is accurate enough

for a reference work.)

131

References

[1] Andrews, G.E.: Partitions, Durfee Symbols, and the Moments of Ranks. Invent.

Math. 169 (2007), 37-73

[2] A. O. L. Atkin and F. Garvan: Relations between the ranks and cranks of partitions,

Ramanujan J., 7 (2003), 343366

[3] Atkin, A.O.L., Hussain, S.M.: Some Properties of Partitions. Trans. Amer. Math.

Soc. 89(1), 184-200 (1958)

[4] Atkin, A.O.L, Swinnerton-Dyer, P.: Some Properties of Partitions. Proc. London

Math. Soc. 3 (4) 84-106 (1954)

[5] Bessenrodt, C.: A bijection for Lebesgues partition identity in the spirit of Sylvester.

Discrete Math. 132(13), 110 (1994)

[6] N.J. Fine.: Basic Hypergeometric Series and Applications. American Mathematical

Society, Providence, RI, 1988

[7] Lewis, R.: On the Ranks of Partitions Modulo 9. Bull. Lond. Math. Soc. 23 417-421

(1991)

[8] O’Brien, J.N.: Some Properties of Partitions with Special Reference to Primes other

than 5, 7, and 11. University College, Durham (1965)

[9] Pak, Igor.: Partition Bijections, a Survey. The Ramanujan Journal, 2006, Vol. 12,

pp. 5-75.

132

[10] Pak, I., Postnikov, A.: A generalization of Sylvesters identity. Discrete Math.

178(13), 277-281 (1998)

[11] Santa-Gadea, N.: On Some Relations for the Rank Moduli 9 and 12. Journal of

Number Theory 40 (2), 130-145 (1992)

[12] N. J. A. Sloane: The On-Line Encyclopedia of Integer Sequences, published elec-

tronically at www.research.att.com/ njas/sequences/ (2006)

[13] Stockhofe, D.: Bijektive Abbildungen auf der Menge der Partitionen einer

naturlichen Zahl. Bayreuth. Math. Schr. (10), 1-59 (1982)

[14] Sylvester, J.J., Franklin, F.: A constructive theory of partitions, arranged in three

acts, an interact and an exodion. Amer. J. Math. 5, 251-330 (1882)

[15] Kim, D., Yee, A.J.: A note on partitions into distinct parts and odd parts. Ramanu-

jan J. 3(2), 227231 (1999)

[16] Zeng, J.: The q-variations of Sylvester’s bijection between odd and strict partitions.

Ramanujan J. 9(3), 289-303 (2005)

Vita

Education

Fall ’99 - Summer ’07 Pennsylvania State University: Graduate Study

Degree expected: Ph.D., Summer 2007

Advisor: Prof. George Andrews

Fall ’95 - Summer ’99 University of Texas at Austin: Undergraduate Study

B.Sc., Mathematics

B.Sc., Physics

Employment and Teaching Experience

Fall ’99 - Summer ’07 Pennsylvania State University, graduate assistant

Contributed Talks

Partitions Seminar, Penn State, Feb ’04

Fibonacci-Like Sequences Among Hyperbinary Partitions

Graduate Student Combinatorics Conference, Univ. of Wisc., April 2006

Lecture Hall Partitions Generalized Toward Glaisher

Midwest Number Theory Conf. for Grad. Students IV, Univ. of Ill., Oct 2006

Congruences for the Full Rank

Graduate Student Combinatorics Conference, Univ. of Wash., April ’07

Flat Partitions, Permutation Descents, and Classical Partition Identities