csm week 1: introductory cross-disciplinary seminar

CSM Week 1: Introductory Cross-Disciplinary Seminar

Combinatorial Enumeration

Dave WagnerUniversity of Waterloo




I. The Lagrange Implicit Function Theorem and Exponential Generating Functions




I. The Lagrange Implicit Function Theorem and Exponential Generating Functions

II. A Smorgasbord of Combinatorial Identities

I. LIFT and Exponential Generating Functions

1. The Lagrange Implicit Function Theorem



2. Exponential Generating Functions




3. There are rooted trees (two ways)1nn




3. There are rooted trees (two ways)

4. Combinatorial proof (sketch) of LIFT

1nn






5. Nested set systems

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5. Nested set systems

6. Multivariate Lagrange

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The human mind has never invented a labor-saving device equal to algebra.

-- J. Willard Gibbs


K: a commutative ring that contains the rational numbers.

F(u) and G(u): formal power series in K[[u]]:

Assume that

n

nnufuF

0

)( n

nnuguG

0

)(

.00 g


K: a commutative ring that contains the rational numbers.

F(u) and G(u): formal power series in K[[u]]:

Assume that

(a) There is a unique formal power series R(x) in K[[x]]

such that

n

nnufuF

0

)( n

nnuguG

0

)(

.00 g

)).(()( xRxGxR


(b) For this formal power series with

the constant term is zero:

))(()( xRxGxR

0

)(n

nnxrxR .00 r


(b) For this formal power series with

the constant term is zero:

For all n>=1 the coefficient of x^n in

F(R(x)) is

))(()( xRxGxR

0

)(n

nnxrxR .00 r

.)()(][1

))((][ 1 nnn uGuFun

xRFx


Proofs:

(i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of

convergence]


Proofs:


convergence]

(ii) Algebraic (formal calculus, formal residue operator)

[requires g_0 to be invertible in K]


Proofs:


convergence]



(iii) Combinatorial (bijective correspondence).)()(][1

))((][ 1 nnn uGuFun

xRFx


Proofs:


convergence]



(iii) Combinatorial (bijective correspondence).)()(][))((][ 1 nnn uGuFuxRFxn


Proofs:


convergence]



(iii) Combinatorial (bijective correspondence) .)()(][!))((][! 1 nnn uGuFunxRFxnn


A class of structures associates to each

finite set another finite set -- this is

the set of A-type structures supported on the set X.

A

X XA





Simplified notation:

A

X XA

.},...,2,1{ nn AA





Simplified notation:

Exponential generating function:

A

X XA

.},...,2,1{ nn AA

!

#)(0 n

xxA

n

nn

A


Minimal requirements on a class of structures:

* depends only on

* If then

XA# X#

YX YX AA


Example: the class of (simple) graphs

is the set of graphs with vertex-set

Exponential generating function

(no particularly useful formula)

G

XG X

2

#

2X

XG#

!2)(

0

2

n

xxG

n

n

n


Example: the class of endofunctions

is the set of all functions


(no particularly useful formula)

X XX :

XX X ###

!)(

0 n

xnx

n

n

n


Example: the class of permutations

is the set of permutations on the set


S

XS X

)!(# XX S#

xn

xnxS

n

n

1

1

!!)(

0


Example: the class of cyclic permutations

is the set of cyclic perm.s on the set


C

XC X

XX

XX )!1(#

0C#

xn

x

n

xnxC

n

nn

n 1

1log

!)!1()(

11


Example: the class of (finite) sets (“ensembles”)

is the set of ways in which is a set.


E

}{XX E X

1XE#

)exp(!

1)(1

xn

xxE

n

n


Example: the class of sets of size k

is the set of ways in which

is a k-element set.


Especially important: the case k=1 of singletons….

has exp.gen.fn x.

)(kE

kX

kXXkX #

#}{)(E X

!)()(

k

xxE

kk

)1(EX


Notice that

xx 1

1logexp

1

1


Notice that

That is… )).(()( xCExS

xx 1

1logexp

1

1


Notice that

That is…

This suggests a relation among classes: E[C].S

)).(()( xCExS

xx 1

1logexp

1

1


Notice that

That is…

This suggests a relation among classes:

A permutation is equivalent to a (finite unordered)set of (pairwise disjoint) cyclic permutations.

E[C].S

)).(()( xCExS

xx 1

1logexp

1

1


X


A permutation is equivalent to a (finite unordered)set of (pairwise disjoint) cyclic permutations.

X


X

xx 1

1logexp

1

1


An endofunction is equivalent to a set of disjointconnected endofunctions.

X


A connected endofunction is equivalent to a cyclic permutation of rooted trees.

X


A rooted tree is equivalent to a root vertex and aset of disjoint rooted (sub-)trees.

X


The Exponential/Logarithmic Formula

For classes A and B,

If every B-structure can be decomposed uniquely as a

finite set of pairwise disjoint A-structures, then

)(exp)( xAxB


The Exponential/Logarithmic Formula

For classes A and B,


finite set of pairwise disjoint A-structures, then

and hence

)(exp)( xAxB

)(log)( xBxA


Example: Let Q be the class of connected graphs.


Example: Let Q be the class of connected graphs.

Since it follows that

!2log)(

0

2

n

xxQ

n

n

n

E[Q]G


Example: Let Q be the class of connected graphs.Since it follows that

More informatively,

records the number of edges in the exponent of y.

!2log)(

0

2

n

xxQ

n

n

n

0

2

0

)(#

!)1(log

!),(

n

nnn

n Q

E

n

xy

n

xyyxQ

n

E[Q]G


The Compositional Formula

For classes A, B, and J:


finite set Y of pairwise disjoint A-structures, together

with a J-structure on Y, then )()( xAJxB


Example: Let K be the class of connectedendofunctions. Let R be the class of rooted trees.



Since it follows that C[R]K

)(1

1log))(()(

xRxRCxK





)(1

1)(exp)(

xRxKx

C[R]K

E[K]

)(1

1log))(()(

xRxRCxK


Sum of classes A and B

An structure on X iseither a red A-structure or a green B-structure on X.

X

-BA


An structure on X

X

-RS


Sum of classes A and BThe exp.gen.fn of is

X

BA

)()( xBxA


Product of classes A and B

An structure on X is an A-structure on Sand a B-structure on X\S (for some subset S of X).

X

-BA*

S SX \


An structure on X

X

-RS*

S SX \


Product of classes A and BThe exp.gen.fn of is

X

BA*

S SX \

)()( xBxA

3. Counting Rooted Trees

A rooted tree is equivalent to a root vertex and aset of disjoint rooted (sub-)trees.

X


X

E[R] XR *


E[R] XR *

)(exp)( xRxxR

From

we deduce that


E[R] XR *

)(exp)( xRxxR

From

we deduce that

LIFT applies with F(u)=u and G(u)=exp(u):


!!

)(][

1)exp(][

1)(][

1

0

11

n

n

k

nuu

nuu

nxRx

n

k

knnnn

E[R] XR *

)(exp)( xRxxR

From

we deduce that



!!

)(][

1)exp(][

1)(][

1

0

11

n

n

k

nuu

nuu

nxRx

n

k

knnnn

E[R] XR *

)(exp)( xRxxR

From

we deduce that


Therefore .)(][!# 1 nnn nxRxnR

5. Nested Set Systems

A nested set system is a pair

in which X is a finite set and is a set of subsets of X

such that

if and then either

or or .

Let N be the class of nested set systems. What is #N_n?

),( X

A B

BA AB BA


X

A nested set system with vertex-set X.



We’ll use the bivariate generating function

!),(

0 ),(

#

n

xyyxN

n

n X n

N



We’ll use the bivariate generating function

This is an exp.gen.fn in the indeterminate x

and records in the exponent of y.

!),(

0 ),(

#

n

xyyxN

n

n X n

N

#


X

A nested set system is proper if it does not containany sets of size zero or one.

Let M be the class of proper nested set systems


X

A proper nested set system


X

),)1(()1(),( yxyMyyxN


X

A proper nested set system is equivalent to a set of disjoint

blobs – each blob is a singleton or a “cell”.


X

A cell is a proper nested set systemfor which --

Let Q be the class of cells.

),( XX


X

A proper nested set system is equivalent to a set of disjoint

blobs – each blob is a singleton or a “cell”.


X

Q]E[XM


X

Q]E[XM

),(exp),( yxQxyxM

The “protoplasm” of a cell is a proper nested set system

that is not empty, not a singleton, and not a cell.


X


X

QXE\MQ )0(


X

QXE\MQ )0(

),(1),(),( yxQxyxMyyxQ


),)1(()1(),( yxyMyyxN

QxM exp

yQyxyyMQ


),)1(()1(),( yxyMyyxN

QxM exp

yQyxyyMQ

xMy

yQ

11


),)1(()1(),( yxyMyyxN

QxM exp

yQyxyyMQ

xMy

yQ

11

xM

y

yxM 11

exp


xM

y

yxM 11

exp


xM

y

yxM 11

exp

y

yM

y

yx

y

y

y

yM

1exp

1exp

11


xM

y

yxM 11

exp

y

yM

y

yx

y

y

y

yM

1exp

1exp

11

Let and

y

yx

y

yz

1exp

1y

yMR

1


)exp(RzR


)exp(RzR

1

1

!k

kk

k

zkR


)exp(RzR

1

1

!k

kk

k

zkR

k

k

k

y

yx

y

y

k

k

y

yM

1exp

1!

1

1

1


)exp(RzR

1

1

!k

kk

k

zkR

k

k

k

y

yx

y

y

k

k

y

yM

1exp

1!

1

1

1

k

k

k

y

yxy

y

y

k

k

y

yyxN

1

)1(exp

1!

)1(),(

1

12


2

1exp

2!4)1,(

1

1

xkk

kxN

kk

k


2

1exp

2!4)1,(

1

1

xkk

kxN

kk

k

1

2/

1

2!4)1,(][!#k

kk

knn

n ek

kxNxnN

Therefore, the number of nested set systems on the

vertex-set {1,2,…,n} is

• n, ~ #N_n (up to k = 500) (k = 500 term of the series)

• 0, 2.000000000000000000000000000000000000000, .4083243888661365954428680604080286918931e-46• 1, 3.999999999999999999999999999999999999997, .2041621944330682977214340302040143459465e-43• 2, 16.00000000000000000000000000000000000000, .1020810972165341488607170151020071729732e-40• 3, 127.9999999999999999999999999999999999998, .5104054860826707443035850755100358648663e-38• 4, 1663.999999999999999999999999999999999951, .2552027430413353721517925377550179324331e-35• 5, 30207.99999999999999999999999999999997499, .1276013715206676860758962688775089662165e-32• 6, 704511.9999999999999999999999999999873828, .6380068576033384303794813443875448310827e-30• 7, 20074495.99999999999999999999999999361776, .3190034288016692151897406721937724155414e-27• 8, 675872767.9999999999999999999999967706124, .1595017144008346075948703360968862077707e-24• 9, 26253131775.99999999999999999999836574318, .7975085720041730379743516804844310388536e-22• 10, 1155636527103.999999999999999999172874000, .3987542860020865189871758402422155194268e-19• 11, 56851643236351.99999999999999958132597320, .1993771430010432594935879201211077597134e-16• 12, 3091106738733055.999999999999788049473799, .9968857150052162974679396006055387985669e-14• 13, 184069292705185791.9999999998926879966856, .4984428575026081487339698003027693992835e-11• 14, 11913835525552734207.99999994566012222874, .2492214287513040743669849001513846996417e-8• 15, 832795579840760643583.9999724801012619416, .1246107143756520371834924500756923498209e-5• 16, 62525006404716521848831.98606091920623040, .6230535718782601859174622503784617491043e-3• 17, 5017971241212451282223096.938744559857556, .3115267859391300929587311251892308745523• 18, 428697615765805738749850118.4015658433400, 155.7633929695650464793655625946154372761• 19, 38844089835957753021198521986.64691355734, 77881.69648478252323968278129730771863803


References

A. JoyalUne theorie combinatoire des series formellesAdv. in Math. 42 (1981), 1-82.

I.P. Goulden, D.M. Jackson“Combinatorial Enumeration”John Wiley & Sons, New York, 1983.

F. Bergeron, G. Labelle, P. Leroux“Combinatorial Species and Tree-like Structures”Cambridge U.P., Cambridge, 1998.

R.P. Stanley“Enumerative Combinatorics, volume II”Cambridge U.P., Cambridge, 1999.

csm week 1: introductory cross-disciplinary seminar

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