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Generalized Inversion Sequences

Carla D. Savage

Department of Computer ScienceNorth Carolina State University

CanaDAM 2013Memorial University of Newfoundland, June 10, 2013

Permutations and Descents

Sn: set of permutations π : {1,2, . . . ,n} → {1,2, . . . ,n}

Desπ: {i ∈ {1, . . .n − 1} | π(i) > π(i + 1)} (descents)

desπ: |Desπ|, the number of descents.

π ∈ S3 Desπ desπ1 2 3 { } 01 3 2 {2} 12 1 3 {1} 12 3 1 {2} 13 1 2 {1} 13 2 1 {1,2} 2

Descent polynomial:

En(x) =∑π∈Sn

xdesπ

E3(x) = 1 + 4x + x2

Eulerian polynomials: En(x)

π ∈ S3 Desπ desπ1 2 3 { } 01 3 2 {2} 12 1 3 {1} 12 3 1 {2} 13 1 2 {1} 13 2 1 {1,2} 2

Descent polynomial:

En(x) =∑π∈Sn

xdesπ

E3(x) = 1 + 4x + x2

π ∈ S3 Desπ desπ1 2 3 { } 01 3 2 {2} 12 1 3 {1} 12 3 1 {2} 13 1 2 {1} 13 2 1 {1,2} 2

Descent polynomial:

En(x) =∑π∈Sn

xdesπ

E3(x) = 1 + 4x + x2

The Eulerian polynomials, En(x)

En(x) =∑π∈Sn

xdesπ

∑t≥0

(t + 1)nx t =En(x)

(1− x)n+1

∑n≥0

En(x)zn

(1− x)

ez(x−1) − x

Inversion Sequences

In = {(e1, . . .en) ∈ Zn | 0 ≤ ei < i}

Encode permutations as inversion sequences φ : Sn → Inφ(π) = (e1, . . . ,en), where

ej = |{i | i < j and π(i) > π(j)}| .

Example:φ(4 3 6 5 1 2) = (0,1,0,1,4,4).

Claim: φ is a bijection with Desπ = Ascφ(π). What is “Asc ”?

Inversion Sequences

In = {(e1, . . .en) ∈ Zn | 0 ≤ ei < i}

ej = |{i | i < j and π(i) > π(j)}| .

Example:φ(4 3 6 5 1 2) = (0,1,0,1,4,4).

Claim: φ is a bijection with Desπ = Ascφ(π).

What is “Asc ”?

Inversion Sequences

In = {(e1, . . .en) ∈ Zn | 0 ≤ ei < i}

ej = |{i | i < j and π(i) > π(j)}| .

Example:φ(4 3 6 5 1 2) = (0,1,0,1,4,4).

Claim: φ is a bijection with Desπ = Ascφ(π). What is “Asc ”?

Ascents

What is an “ascent” in an inversion sequence?

ei < ei+1?

i + 1?

If 0 ≤ ej < j for all j ≤ n, then for 1 ≤ i < n,

ei < ei+1 iffei

i + 1.

Ascents

ei < ei+1? orei

i + 1?

ei < ei+1 iffei

i + 1.

Ascents

ei < ei+1? orei

i + 1?

ei < ei+1 iffei

i + 1.

View inversion sequences as lattice pointsin a (half-open) 1 × 2 × · · · × n box

View ascent constraints as hyperplane constraints:

0 < e1 andei

i + 1, 1 ≤ i < n

π ∈ S3 e ∈ In Asc e1 2 3 (0,0,0) { }1 3 2 (0,0,1) {2}2 1 3 (0,1,0) {1}2 3 1 (0,0,2) {2}3 1 2 (0,1,1) {1}3 2 1 (0,1,2) {1,2}

s-inversion sequencesFor any sequence s = (s1, s2, . . . , sn) of positive integers:

I(s)n = {(e1, . . . ,en) ∈ Zn | 0 ≤ ei < si for 1 ≤ i ≤ n} .

(lattice points in a half-open s1 × s2 × · · · × sn box)

∣∣∣I(s)n

∣∣∣ = s1s2 · · · sn

%pause

An ascent of e is a position i :1 ≤ i < n and

If e1 > 0 then 0 is an ascent.

%pause

Example: (2,4,5) ∈ I(3,5,7)n

Asc e = {0,1}

2 6∈ Asc e since 4/5 6< 5/7

Ascent polynomials of s-inversion sequences

A(s)n (x) =

∑e∈I(s)

xasc e

(2,4)-inversion sequences

A(2,4)n (x) = 1 + 6x + x2

Ascent sets:{ } yellow dot{0} blue square{1} red diamond{0,1} black dot

A(3,5)n (x) = 1 + 10x + 4x2

Ascent polynomials of s-inversion sequences

A(s)n (x) =

∑e∈I(s)

xasc e

A(2,4)n (x) = 1 + 6x + x2

A(3,5)n (x) = 1 + 10x + 4x2

Call A(s)n (x) the s-Eulerian polynomial since,

when s = (1,2, . . . ,n),

A(s)n (x) =

∑e∈I(s)

x asc e =∑π∈Sn

xdesπ = En(x),

the Eulerian polynomial.

(Recall bijection φ : Sn → In with Desπ = Ascφ(π))

Why s-inversion sequences?

• Natural model for combinatorial structures• Can prove general properties of the s-Eulerian polynomials• Surprising results follow using Ehrhart theory• Can be encoded as lecture hall partitions• Lead to a natural refinement of the s-Eulerian polynomials• Help answer questions about lecture hall partitions

sequence s s-Eulerian polynomial

(1,2,3,4,5,6) 1 + 57 x + 302 x2 + 302 x3 + 57 x4 + x5

(2,4,6,8,10) 1 + 237 x + 1682 x2 + 1682 x3 + 237 x4 + x5

(6,5,4,3,2,1) 1 + 57 x + 302 x2 + 302 x3 + 57 x4 + x5

(1,1,3,2,5,3) 1 + 20 x + 48 x2 + 20 x3 + x4

(1,3,5,7,9,11) 1 + 358 x + 3580 x2 + 5168 x3 + 1328 x4 + 32 x5.

(7,2,3,5,4,6) 1 + 71 x + 948 x2 + 2450 x3 + 1411 x4 + 159 x5

Signed permutationsand

(2,4,6, . . . ,2n)-inversion sequences

∣∣∣I(2,4,6,...,2n)n

∣∣∣ = 2nn!

Signed Permutations Bn

Bn = {(σ1, . . . , σn) | ∃π ∈ Sn, ∀i σi = ±π(i)}

π ∈ Bn Desπ( 1, 2) { }(-1, 2) {0}( 1,-2) {1}(-1,-2) {0,1}( 2, 1) {1}(-2, 1) {0}( 2,-1) {1}(-2,-1) {0}

descent polynomial:

1 + 6x + x2

Desσ = {i ∈ {0, . . . ,n−1} | σi > σi+1},

with the convention that σ0 = 0.

Signed Permutations Bn

Bn = {(σ1, . . . , σn) | ∃π ∈ Sn, ∀i σi = ±π(i)}

π ∈ Bn Desπ( 1, 2) { }(-1, 2) {0}( 1,-2) {1}(-1,-2) {0,1}( 2, 1) {1}(-2, 1) {0}( 2,-1) {1}(-2,-1) {0}

descent polynomial:

1 + 6x + x2

ascent polynomial:

1 + 6x + x2

Theorem (Pensyl, S 2012/13)

∑σ∈Bn

xdesσ = A(2,4,...,2n)n (x).

Proof.

There is a bijection Θ : Bn → I(2,4,...,2n)n satisfying

Desσ = Asc Θ(σ).

Theorem (Pensyl, S 2012/13)

∑σ∈Bn

xdesσ = A(2,4,...,2n)n (x).

Proof.

There is a bijection Θ : Bn → I(2,4,...,2n)n satisfying

Desσ = Asc Θ(σ).

(s1, s2, . . . , sn)-inversion sequencesvs.

(sn, sn−1, . . . , s1)-inversion sequences

Example from table:

A(1,2,3,4,5,6)6 (x) = 1 + 57 x + 302 x2 + 302 x3 + 57 x4 + x5

= A(6,5,4,3,2,1)6 (x)

Theorem (S, Schuster 2012; Liu, Stanley 2012)

For any sequence (s1, s2, . . . , sn) of positive integers,

A(s1,s2,...,sn)n (x) = A(sn,sn−1,...,s1)

n (x).

Reversing s preserves the ascent polynomial

1 + 10 x + 4 x2

but not necessarily the partition into ascent sets

3.Roots of s-Eulerian polynomials

Example from table:

A(7,2,3,5,4,6)6 (x) = 1+71 x +948 x2 +2450 x3 +1411 x4 +159 x5

Roots in the intervals:

[[−19/1024,−9/512], [−77/1024,−19/256],

[−423/1024,−211/512], [−1701/1024,−425/256],

[−3435/512,−6869/1024]]

Theorem (S, Visontai 2012)

For every sequence s of positive integers, A(s)n (x) has all real

roots.

Corollary (Frobenius 1910; Brenti 1994)

The descent polynomials for Coxeter groups of types A and Bhave all real roots.

New: ([S, Visontai 2013]) Method can be adapted to type D.

roots.

Corollary

For any s, the sequence of coefficents of the s-Eulerianpolynomial is unimodal and log-concave.

Example A(7,2,3,5,4,6)6 (x):

1, 71, 948, 2450, 1411, 159, 1

roots.

Corollary

For any s, the sequence of coefficents of the s-Eulerianpolynomial is unimodal and log-concave.

Example A(7,2,3,5,4,6)6 (x):

1, 71, 948, 2450, 1411, 159, 1

Lecture Hall Polytopesand

Ehrhart Theory

Lecture hall polytopess-lecture hall polytope:

P(s)n =

{λ ∈ Rn

∣∣∣ 0 ≤ λ1

s1≤ λ2

s2≤ · · · ≤ λn

sn≤ 1

P(3,5)2

P(s)n =

{λ ∈ Rn

∣∣∣ 0 ≤ λ1

s1≤ λ2

s2≤ · · · ≤ λn

sn≤ 1

P(3,5)2

t-th dilation of P(s)n :

tP(s)n = {tλ | λ ∈ P(s)

Ehrhart polynomial of P(s)n :

i(P(s)n , t) = |tP(s)

n ∩ Zn|.

P(s)n =

{λ ∈ Rn

∣∣∣ 0 ≤ λ1

s1≤ λ2

s2≤ · · · ≤ λn

sn≤ 1

P(3,5)2

tP(s)n = {tλ | λ ∈ P(s)

i(P(s)n , t) = |tP(s)

n ∩ Zn|.

P(s)n =

{λ ∈ Rn

∣∣∣ 0 ≤ λ1

s1≤ λ2

s2≤ · · · ≤ λn

sn≤ 1

P(3,5)2

tP(s)n = {tλ | λ ∈ P(s)

i(P(s)n , t) = |tP(s)

n ∩ Zn|.

Connection between lecture hall polytopes andinversion sequences

Theorem (S, Schuster 2012)

For any sequence s of positive integers,

∑t≥0

i(P(s)n , t) x t =

∑e∈I(s)

nx asc (e)

(1− x)n+1 .

The sequences

s = (1, 1, 3, 2, 5, 3, 7, 4, . . .)

s = (1, 4, 3, 8, 5, 12, 7, 16, . . .)

Note:∣∣∣I(1,1,3,2,5,3,...,2n−1,n)2n

∣∣∣ = n!(1 · 3 · 5 · · · · · 2n − 1) =(2n)!

A(1,1,3,2,...,2n−1,n)2n is the descent polynomial for permutations of

the multiset {1,1,2,2, . . . ,n,n}.

Bijective proof?

Conjecture (S, Visontai 2012)

A(1,4,3,8,...,2n−1,4n)2n is the descent polynomial for the signed

permutations of {1,1,2,2, . . . ,n,n}.

the multiset {1,1,2,2, . . . ,n,n}.

Bijective proof?

the multiset {1,1,2,2, . . . ,n,n}.

Bijective proof?

The sequences s = (1, k + 1,2k + 1,3k + 1, . . .)

k = 1 : (1, 2, 3, 4, 5, . . .)k = 2 : (1, 3, 5, 7, 9, . . .)

In,k = I (1,k+1,2k+1,...,(n−1)k+1)n

An,k (x) = A(1,k+1,2k+1,...,(n−1)k+1)n (x)

Recall An,1(x) = En(x).

The sequences s = (1, k + 1,2k + 1,3k + 1, . . .)

k = 1 : (1, 2, 3, 4, 5, . . .)k = 2 : (1, 3, 5, 7, 9, . . .)

In,k = I (1,k+1,2k+1,...,(n−1)k+1)n

An,k (x) = A(1,k+1,2k+1,...,(n−1)k+1)n (x)

Recall An,1(x) = En(x).

The 1/k -Eulerian polynomials

Theorem (S, Viswanathan 2012)

For positive integer k,

An,k (x) =∑

e∈In,k

xasc e

∑t≥0

(t − 1 + 1

)(kt + 1)nx t =

An,k (x)

(1− x)n+ 1k∑

An,k (x)zn

(1− x

ekz(x−1) − x

∑e∈In,k

x asc e =∑π∈Sn

x exc πkn−#cyc π,

excπ = |{i | π(i) > i}|

and #cycπ is the number of cycles in the disjoint cyclerepresentation of π.

Combinatorial proof?

∑e∈In,k

x asc e =∑π∈Sn

x exc πkn−#cyc π,

excπ = |{i | π(i) > i}|

and #cycπ is the number of cycles in the disjoint cyclerepresentation of π.

Combinatorial proof?

Lecture Hall Partitions

{λ ∈ Zn | λ1

1≤ λ2

2≤ · · · ≤ λn

Lecture Hall Partitions

{λ ∈ Zn | λ1

1≤ λ2

2≤ · · · ≤ λn

s-lecture hall partitions

L(s)n =

{λ ∈ Zn | λ1

s1≤ λ2

s2≤ · · · ≤ λn

Theorem (Bousquet-Mélou, Eriksson 1997)

For s = (1,2, . . .n),∑λ∈L(s)

q|λ| =1

(1− q)(1− q3) · · · (1− q2n−1),

where |λ| = λ1 + · · ·+ λn.

(What other sequences s give rise to nice generatingfunctions?)

s-lecture hall partitions

L(s)n =

{λ ∈ Zn | λ1

s1≤ λ2

s2≤ · · · ≤ λn

For s = (1,2, . . .n),∑λ∈L(s)

q|λ| =1

(1− q)(1− q3) · · · (1− q2n−1),

where |λ| = λ1 + · · ·+ λn.

(What other sequences s give rise to nice generatingfunctions?)

8.Fundamental Lecture Hall Parallelepiped

s = (3,5)

s = (2,4,6)

Fundamental (half-open) s-lecture hall parallelepiped:

Π(s)n =

ciwi | 0 ≤ ci < 1

where wi = [0, . . . ,0, si , si+1, . . . sn].

Theorem (Liu, Stanley 2012)

There is a bijection between I(s)n and Π

(s)n ∩ Zn.

I(3,5)2

Π(3,5)2 ∩ Z2

I(2,4,6)3

(2,4,6)3 ∩ Z3

I(3,5)2

Π(3,5)2 ∩ Z2

I(2,4,6)3

(2,4,6)3 ∩ Z3

Inflated s-Eulerian polyomials

∑λ∈Π

Define the inflated s-Eulerian polyomial by

Q(s)n (x) =

∑λ∈Π

xλn .

Theorem (Pensyl,S 2013)

For any sequence s of positive integers,

Q(s)n (x) =

∑e∈I(s)

xsnasc e−en

For s = (1,2, . . . ,n), the coefficient sequence of Q(s)n gives an

interesting refinement of the Eulerian numbers:

Coefficient sequence:

1, 1, 2, 4, 4, 4, 4, 2, 1, 1

For s = (1,3, . . . ,2n − 1), the coefficient sequence of Q(s)n (x) is

symmetric (but not the coefficient sequence of A(s)n (x).)

1, 1, 2, 4, 4, 6, 9, 10, 10, 11, 10, 10, 9, 6, 4, 4, 2, 1, 1

For s = (1,1,2,3,5,8, . . .), the coefficient sequence of Q(s)n is

not symmetric for n ≥ 5.

1, 1, 2, 2, 4, 4, 4, 4, 4, 2, 1, 1,

Gorenstein cones and self-reciprocal generating functions

Self-reciprocal:

satisfies f (q) = qbf (1/q) for some nonnegative integer b

Examples:

1 + x + 2x2 + 4x3 + 4x4 + 4x5 + 4x6 + 2x7 + x8 + x9

1(1− q)(1− q3)(1− q5)

A pointed rational cone C ⊆ Rn is Gorenstein if there exists apoint c in the interior C0 of C such that C0 ∩ Zn = c + (C ∩ Zn).

Theorem (Special case of a result due to Stanley 1978)

The s-lecture hall cone is Gorenstein if and only if Q(s)n (x) is

self-reciprocal; also, if and only if the following is self reciprocal:

f (s)n (q) =

∑λ∈L(s)

Theorem (Bousquet-Mélou, Eriksson 1997; Beck, Braun,Köppe, S, Zafeirakopoulos 2012)

The s-lecture hall cone is Gorenstein if and only if there existsc ∈ Zn satisfying

cjsj−1 = cj−1sj + gcd(sj , sj−1)

for j > 1 with c1 = 1.

Theorem (BBKSZ)

[Beck, Braun, Köppe, S, Zafeirakopoulos 2012]

Let s be a sequence of positive integers defined by

sn = `sn−1 + msn−2, (∗)

with s0 = 0, s1 = 1. Then the s-lecture hall cone in Gorensteinfor all n if and only if m = −1.

Sequences (*) with m = −1 are called `-sequences.

Theorem (BBKSZ)

[Beck, Braun, Köppe, S, Zafeirakopoulos 2012]

Let s be a sequence of positive integers defined by

sn = `sn−1 + msn−2, (∗)

with s0 = 0, s1 = 1. Then the s-lecture hall cone in Gorensteinfor all n if and only if m = −1.

Sequences (*) with m = −1 are called `-sequences.

`-sequences

sn = `sn−1 − sn−2,

with s0 = 1, s1 = 1.

1, 2, 3, 4, 5, 6, 7, 8, 9, . . .

1, 3, 8, 21, 55, 144, 377, 987, 2584, . . .

If s is an `-sequence,∑λ∈L(s)

q|λ| =1

(1− q)(1− qs1+s2)(1− qs2+s3) · · · (1− qsn−1+sn ).

Conversely, by the BBKSZ Theorem, for a sequence of the form(*) unless s is an `-sequence, the s-lecture hall partitionscannot, for all n, have a generating function of the form

1(1− qc1)(1− qc2) · · · (1− qcn )

Question

What combinatorial family is being represented by thes-inversion sequences when s is an `-sequence?

(When ` = 2, the answer is permutations.)

References

1. s-Lecture hall partitions, self-reciprocal polynomials, andGorenstein cones, M. Beck, B, Braun, M. Köppe, C. D. Savage,and Z. Zafeirakopoulos, submitted.

2. Lecture hall partitions, M. Bousquet-Mélou and K. Eriksson,Ramanujan J., 1(1):101-111, 1997.

3. Lecture hall partitions. II, M. Bousquet-Mélou and K.Eriksson, Ramanujan J., 1(2):165-185, 1997.

4. q-Eulerian polynomials arising from Coxeter groups, F.Brenti, European J. Comb., 15:417.441, September 1994.

5. The Lecture Hall Parallelepiped, F. Liu and R. P. Stanley, July2012, arXiv:1207.6850

6. Rational lecture hall polytopes and inflated Eulerianpolynomials, T. W. Pensyl and C. D. Savage, Ramanujan J., Vol.31 (2013) 97-114.

7. Lecture hall partitions and the wreath products Ck o Sn, T. W.Pensyl and C. D. Savage, Integers, 12B, #A10 (2012/13).

8. Ehrhart series of lecture hall polytopes and Eulerianpolynomials for inversion sequences, C. D. Savage and M. J.Schuster, Journal of Combinatorial Theory, Series A, Vol. 119(2012) 850-870.

9. The 1/k-Eulerian polynomials, C. D. Savage and G.Viswanathan, The Electronic Journal of Combinatorics, Vol. 19(2012) Research Paper P9 , 21 pp. (electronic).

10. The Eulerian polynomials of type D have only real roots, C.D. Savage and M. Visontai, FPSAC 2013, Paris.

11. The s-Eulerian polynomials have only real roots, C. D.Savage and M. Visontai, submitted, arXiv:1208.3831, August2012.

12. Hilbert functions of graded algebras, R. P. Stanley,Advances in Math., 28(1):57.83, 1978.

Thank you!

carla d. savage - cms-smccanadam.math.ca/2013/program/slides/savage.carla.inversion_sequ… ·...

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