CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo

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


1. Multivariate Lagrange Implicit Function Theorem



2. The MacMahon Master Theorem




3. Cartier-Foata (Viennot) Heap Inversion





4. Karlin-MacGregor/Lindstrom/Gessel-Viennot






5. Kirchhoff’s Matrix Tree Theorem







6. The “Four-Fermion Forest Theorem” (C-J-S-S-S)

1. Multivariate LIFT

Commutative ring K

Indeterminates and

Power series in K[[u]].

),...,,( 21 nuuuu ),...,,( 21 nxxxx

)(),...,(),( 1 uuu nGGF


Commutative ring K

Indeterminates and

Power series in K[[u]].

(a) There are unique power series in K[[x]]

such that for each 1 <= j <= n.

),...,,( 21 nuuuu ),...,,( 21 nxxxx

)(),...,(),( 1 uuu nGGF

)(xjR

),...,( 1 njjj RRGxR


(b) For these power series and for any monomial

(I.J. Good, 1960)

j

i

i

jijn u

G

G

uFRRF

)(

)(det)()(][),...,(][ 1

u

uuGuux ααα

n

nxxx ...21

21αx

)(xjR


Special case of Multivariate LIFT in which each

is a homogeneous linear form.

ninii ucucG ...)( 11u


Special case of Multivariate LIFT in which each

is a homogeneous linear form.

(MacMahon, 1915)

ninii ucucG ...)( 11u

ijiij

n

inini cxxcxc

i

det

1][...][

111

αα xx


This can be rephrased as….

ijij

n

inini cxcxc

i

det

1...][

111

0α

αx



The matrix represents an endomorphism

on an n-dimensional vector space V.

ijij

n

inini cxcxc

i

det

1...][

111

0α

αx

)( ijcC



The matrix represents an endomorphism

on an n-dimensional vector space V.

There are induced endomorphisms on the symmetric

powers of V, and on the exterior powers of V.

ijij

n

inini cxcxc

i

det

1...][

111

0α

αx

)( ijcC

CS m

Cm


The traces of these induced endomorphisms satisfy

m

n

inini

mi

xcxcCtrS|| 1

11 ...][α

αx


The traces of these induced endomorphisms satisfy

m

n

inini

mi

xcxcCtrS|| 1

11 ...][α

αx

mmn

m

mijij TCtrTc

0

)1(det


By the MacMahon Master Theorem…

This is called the “Boson-Fermion Correspondence”

1

00

)1(

mm

n

m

m

m

mm TCtrTCtrS


By the MacMahon Master Theorem…

This is called the “Boson-Fermion Correspondence”

(Garoufalidis-Le-Zeilberger, 2006)“quantum” MacMahon Master Theorem.

1

00

)1(

mm

n

m

m

m

mm TCtrTCtrS

3. Cartier-Foata/Viennot Heap Inversion

Another example of the Boson-Fermion Correspondence

arising from symmetric functions….

Countably many indeterminates,...),( 21 xxx




Countably many indeterminates

Elementary symmetric functions

k

kiii

iiik xxxe...21

21...)(x

,...),( 21 xxx




Countably many indeterminates

Elementary symmetric functions

Complete symmetric functions

k

kiii

iiik xxxe...21

21...)(x

k

kiii

iiik xxxh...21

21...)(x

,...),( 21 xxx


Generating functions…

10

1)()(i

ik

kk TxTeTE x



10

1)()(i

ik

kk TxTeTE x

10 1

1)()(

i ik

kk Tx

ThTH x



Clearly

10

1)()(i

ik

kk TxTeTE x

10 1

1)()(

i ik

kk Tx

ThTH x

)(

1)(

TETH


Let G=(V,E) be a simple graph.A subset S of V is stable provided that no edge

of G has both ends in S.


Let G=(V,E) be a simple graph.A subset S of V is stable provided that no edge

of G has both ends in S.

Introduce indeterminates

The stable set enumerator of G is

}:{ Vvxv x

)(

);(stableS

SGZ xx


Let G=(V,E) be a simple graph.A subset S of V is stable provided that no edge of

G has both ends in S.


The stable set enumerator of G is

(Partition function of a zero-temperature lattice gas on G with repulsive nearest-neighbour interactions.)

}:{ Vvxv x

)(

);(stableS

SGZ xx


Let G=(V,E) be a simple graph.


Say that these commute only for non-adjacent vertices:

if and only if

}:{ Vvxv x

vwwv xxxx Ewv },{


Let G=(V,E) be a simple graph.


Say that these commute only for non-adjacent vertices:

if and only if

Let be the set of all finite strings of vertices, modulo the equivalence relation generated by these commutation relations.

}:{ Vvxv x

vwwv xxxx Ewv },{

/*V


(Cartier-Foata, 1969)

This identity is valid for power series with merely partially commutative indeterminates, as above.

/* );(

1

V GZ

xx


(Cartier-Foata, 1969)

This identity is valid for power series with merely partially commutative indeterminates, as above.

(There are several variations and generalizations of this.)

(Viennot, 1986)(Krattenthaler, preprint)

/* );(

1

V GZ

xx


Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane.



Let each edge e be weighted by a value w(e) in some commutative ring K.




For a path P, let w(P) be the product of the weights of the edges of P.





Fix vertices in that cyclic order around the boundary of the infinite face of G.

1121 ,...,,,...,, ZZZAAA kkk





Fix vertices in that cyclic order around the boundary of the infinite face of G.

Let be the generating function for

all (directed) paths from A_i to Z_j.

1121 ,...,,,...,, ZZZAAA kkk

ji ZAP

ji PwZAM:

)(),(


1A

2A

kA

1Z

1kZ

kZ


The generating function for the set of all k-tuples of paths

such that* the paths P_i are internally vertex-disjoint* each P_i goes from A_i to Z_iis

),...,,( 21 kPPP

),(det)()...()(),...,,(

21

21

jiPPP

k ZAMPwPwPwk


The generating function for the set of all k-tuples of paths

such that* the paths P_i are internally vertex-disjoint* each P_i goes from A_i to Z_iis

),...,,( 21 kPPP

),(det)()...()(),...,,(

21

21

jiPPP

k ZAMPwPwPwk


Application:

vertical edges get weight 1.

horizontal edges (a,b)—(a+1,b) getweight x_b

1

2

3

4

5

6

7

8

x

x

x

x

x

x

x

x


1

2

3

4

5

6

7

8

x

x

x

x

x

x

x

x

)1,(a

),( ka

The generating functionfor all paths from to

is a complete symmetricfunction

)1,(a ),( ka

)(xkh


1

2

3

4

5

6

7

8

x

x

x

x

x

x

x

x

)1,(a

),( ka

The generating functionfor all paths from to

is a complete symmetricfunction

The path shown is codedby the sequence2 2 4 7 7

)1,(a ),( ka

)(xkh


1

2

3

4

5

6

7

8

x

x

x

x

x

x

x

x

Sets of vertex-disjointpaths are encoded bytableaux : 1 1 3 62 2 4 7 73 5 5 8


1

2

3

4

5

6

7

8

x

x

x

x

x

x

x

x

Sets of vertex-disjointpaths are encoded bytableaux : 1 1 3 62 2 4 7 73 5 5 8

The generating function fortableaux of a given shapeis a symmetric function…

skew Schur function )(/ xs


(dual) Jacobi-Trudy Formula

* When these correspond to the irreducible

representations of the symmetric groups.

)(det)(/ xxji jihs


(dual) Jacobi-Trudy Formula

* When these correspond to the irreducible

representations of the symmetric groups.

* They are the minors of “generic” Toeplitz matrices.

)(det)(/ xxji jihs


Let G=(V,E) be a finite connected (multi-)graph.



Direct each edge e with ends v and w arbitrarily:

Either v—ew or w—ev.



Direct each edge e with ends v and w arbitrarily:

Either v—ew or w—ev.

Define a signed incidence matrix of G to be theV-by-E matrix D with entries

otherwise

ev

ve

Dve

0

1

1


Fix indeterminates

}:{ Eeye y


Fix indeterminates

Let Y be the E-by-E diagonal matrix

}:{ Eeye y

):( EeydiagY e


Fix indeterminates


The weighted Laplacian matrix of G is

}:{ Eeye y

):( EeydiagY e

*DYDL


A graph


:= G

-1 0 0 -1 -1 0 0 0 0 -1 -1

1 0 -1 0 0 -1 0 0 0 0 0

0 0 0 0 0 1 -1 0 -1 0 1

0 -1 1 1 0 0 0 0 0 0 0

0 0 0 0 1 0 0 -1 1 0 0

0 1 0 0 0 0 1 1 0 1 0

A signed incidence matrix for it


y1 y4 y5 y10 y11 y1

y11 y4

y5 y10

y1 y1 y3 y6

y6 y3 0 0

y11 y6

y6 y7 y9 y11 0 y9 y7

y4 y3 0 y2 y3 y4 0 y2

y5 0 y9 0 y5 y8 y9 y8

y10 0 y7 y2

y8 y2 y7 y8 y10

Its weighted Laplacian matrix


Fix indeterminates



Fix any “ground vertex”

}:{ Eeye y

):( EeydiagY e

*DYDL

Vv 0


Fix indeterminates



Fix any “ground vertex”

Let be the submatrix of L obtained by deleting the row and the column indexed by

}:{ Eeye y

):( EeydiagY e

*DYDL

Vv 0

)|( 00 vvL

0v


With the notation above…

where the summation is over the set of all spanning trees of G.

T Te

eyvvL )|(det 00



where the summation is over the set of all spanning trees of G.

Proof uses the Binet-Cauchy determinant identity and…

T Te

eyvvL )|(det 00


Key Lemma:

Let and with ES VR )(#)(#)(# VRS


Key Lemma:

Let and with

Let M be the square submatrix of D obtained by* deleting rows indexed by vertices in R, and* keeping only columns indexed by edges in S.

ES VR )(#)(#)(# VRS

R

S

M


Key Lemma:

Let and with

Let M be the square submatrix of L obtained by* deleting rows indexed by vertices in R, and* keeping only columns indexed by edges in S.

Then if (V,S) is a forest in which each tree has exactly one vertex in R,

and otherwise

ES VR )(#)(#)(# VRS

1det M

0det M



where the summation is over the set of all spanning forests F of G such that each

component of F contains exactly one vertex in R.

“Shorthand” notation:

F

FRRL y)|(det

Fe

eF yy



where the summation is over the set of all spanning forests F of G

F

F FintreeT

TVLI y

:

)(#)det(



where the summation is over the set of all spanning forests F of G

But… we really want a formula without the multiplicities on the RHS….

F

F FintreeT

TVLI y

:

)(#)det(

???F

Fy

6. The “Four-Fermion Forest Theorem” of C-J-S-S-S

Caracciolo-Jacobsen-Saleur-Sokal-Sportiello (2004)

The generating function for spanning forests of G is

Eijejjiie

F

F yLId ψθψθy )(exp)(


“Shorthand” notation

The greek letters stand for fermionic (anticommuting)

variables. et

cetera

in particular

nnddddddd ...)( 2211ψθ

ijji

02 i



The greek letters stand for fermionic (anticommuting)

variables.

is an operator – it means keep track only of terms in which each variable occurs exactly once, counting each such term with an appropriate sign.

nnddddddd ...)( 2211ψθ

)(ψθd


For any square matrix M

)exp()()det( ψθψθ MdM





j

n

i

n

jijimM

1 1

ψθ



Compare with C-J-S-S-S:


Eijejjiie

F

F yLId ψθψθy )(exp)(


“Traditionally” each vertex gets a commuting (bosonic) indeterminate vx


“Traditionally” each vertex gets a commuting (bosonic) indeterminate

In C-J-S-S-S this has two anticommuting (fermionic) “superpartners”

vx

v v




and the boson is “integrated out”

vx

v v





v v





The integral is interpreted combinatorially, some very pretty sign-cancellations occur, and only the forests survive, each exactly once.

v v

I believe there is a department of mind conducted independent of consciousness, where things are fermented and decocted, so that when they are run off they come clear.

-- James Clerk Maxwell

Documents

CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo