Graph Colorings and Related Symmetric Functions

Graph Colorings

And

Symmetric Functions

Ideas and

Applications

Subtitle

Description of

Results

Interesting

Applications

Notable

Op en problems

Richard P Stanley

Department of Mathematics

Massachusetts Institute of Technology

Cambridge MA

email rstanmathmitedu

version of June

Partially supp orted by NSF grant DMS

Schur p ositivity

Let G b e a nite graph with no lo ops edges from a vertex to itself or multiple

edges In we dened a symmetric function X X x x which

G G

generalizes the chromatic p olynomial n of G In this pap er we will rep ort

on further work related to this symmetric function

We rst review the denition of X We will denote by V fv v g

G d

the vertex set and by E the edge set of G A coloring of G is any function

V P f g If is a coloring then set

x x

v V

where x x are commuting indeterminates We say that the coloring is

proper if there are no mono chromatic edges ie if uv E then u v

Dene

X X x x

G G

summed over all prop er colorings Thus X is a homogeneous symmet

ric function of degree d in the variables x x x Moreover it is

immediate from the denition of X that

X n

G G

where in general for a symmetric function f we denote by f the substi

tution x x x x x

n n n

The basic prop erties of the symmetric function X are discussed in

In particular we considered the expansion of X in terms of the four bases

m the monomial symmetric functions p the p ower sum symmetric func

tions s the Schur functions and e the elementary symmetric functions

We are assuming a basic knowledge of symmetric functions such as may b e

found in Chapter I of One of the most interesting op en problems con

cerning X is the following A subp oset Q of a p oset partially ordered set

P is said to b e induced if whenever u v Q and u v in P then u v in Q

A nite p oset P is said to b e free if it contains no induced subp oset

isomorphic to the disjoint union of a threeelement chain and a oneelement

chain We denote the incomparability graph of a p oset P by incP If b is

a symmetric function basis then we say that the graph G is bpositive if the

expansion of X in the basis b has nonnegative co ecients

Conjecture Conj If P is a free poset then incP

is epositive

This conjecture is true for free p osets ie the edge complement G of

G is bipartite Cor

Although the ab ove conjecture remains op en the weaker result that in

comparability graphs of free p osets are sp ositive was proved by V

Gasharov Ch I I Thm as mentioned in Thm In fact

Gasharov gives a combinatorial interpretation of the co ecients which we

now explain stated slightly dierently from Gasharov

Denition Let P b e a nite p oset with d elements A P tableau

of shap e d is a map P P satisfying the following three conditions

a For all i we have i

b is a prop er coloring of incP ie if u v then u v or v u

c By b the elements of the set i form a chain say u u

u Similarly supp ose that the elements of i are v v

v Then for all i and all j we require that v u

i j j

i+1

Note that if P is itself a chain v v then a map P P

is a P tableau of shap e if and only if the sequence v v is a

lattice p ermutation of shap e as dened eg in p Def

Since there is a simple bijection b etween lattice p ermutations of shap e and

standard Young tableaux of shap e p we may regard a P tableaux

of shap e when P is a chain as a standard Young tableau of shap e Hence

for general P a P tableau of shap e should b e regarded as a generalization

of a standard Young tableau of shap e

Let f P denote the number of P tableaux of shap e

Theorem V Gasharov Let P be a free poset and G

incP Then

X f P s

Gasharov proves when P is free by an involution principle

argument Since b oth sides have simple combinatorial interpretations there

should b e a direct bijective pro of When P is a chain the identity b ecomes

f s x x x

where f denotes the number of standard Young tableaux of shap e A bi

jective pro of of this identity is provided precisely by the RobinsonSchensted

corresp ondence so we are seeking a generalization of RobinsonSchensted

Such a generalization can b e gleaned from the work of A Magid x

though a simpler direct bijection would b e desirable

A claw is a complete bipartite graph K A graph is clawfree if no

induced subgraph is a claw Note that K is the incomparability graph

of the disjoint union of a threeelement chain and oneelement chain

and that K is the incomparability graph of no other p oset It follows that

an incomparability graph incP is clawfree if and only if P is free

Thus it is natural to ask whether Conjecture or Theorem extends to

clawfree graphs In Figure we gave an example of a clawfree graph

which isnt ep ositive On the other hand the question of whether clawfree

graphs might b e sp ositive was rst raised by Gasharov unpublished and

there now seems to b e enough evidence to make it into a conjecture

Conjecture If G is clawfree then G is spositive

There is a nice combinatorial consequence of the sp ositivity of a graph

G Recall from that a stable partition of G of type d is a partition of

the vertex set V of G into stable or indep endent subsets of sizes

Dene the graph G to b e nice if whenever there exists a stable partition of

G of type and whenever dominance or ma jorization order called

the natural order in p then there exists a stable partition of G

of type For instance the claw K is not nice since there exists a stable

partition of type but not of type

Prop osition If G is spositive then G is nice

Pro of By denition of X G p ossesses a stable partition of type if

and only if the co ecient of m in X is nonzero see Prop The

pro of now follows from the fact that the co ecient of m in the Schur

function s is nonzero if and only if 2

As a small bit of evidence for Conjecture we have the following result

Prop osition A graph G and al l its induced subgraphs are nice if

and only if G is clawfree

Pro of Since claws are not nice the only if part follows To prove

the if part we use the simple fact that if covers in dominance order

then is obtained from by subtracting from some part and adding

to some part Not all such need b e covered by Hence it

j i

suces to prove that if a clawfree graph H has a stable partition of type

and if is as just describ ed then H has a stable partition of type Let

W b e a subset of V which is the union of a blo ck A of of size and a

blo ck B of size Let H denote the restriction of H to W Hence H is

j W W

bipartite Since H is clawfree every vertex of H has degree one or two so

H is a disjoint union of paths and cycles The vertices of each path and

cycle alternate b etween A and B Since A B there is a comp onent of

H which is a path starting and ending in A Let P denote the vertex set

of this path Replace A and B by A P B P and A P B P

This yields a stable partition of H of type completing the pro of 2

Griggs has made a conjecture Problem equivalent to the statement

that the incomparability graph of the b o olean algebra B is nice This sug

gests that incB might b e sp ositive which is true for n Perhaps

even the incomparability graph of any distributive lattice is sp ositive This

seems quite unlikely however since in particular the distributive lattice L of

Figure has the prop erty that incL f g is not sp ositive though incL

is itself sp ositive The mo dular lattice of Figure has an incomparability

graph which isnt nice and hence isnt sp ositive There is a partition into

chains of type but not

r r r

Figure A distributive lattice L for which incL f g isnt sp ositive

r r

Q AA

Q r r r r r r A

A r

Figure A mo dular lattice whose incomparability graph isnt nice

Ganalogues of symmetric functions

For each graph G we dene a homomorphism from the ring of symmet

ric functions to the p olynomial ring in the vertices of G which is closely

connected with the symmetric function X This homomorphism is closely

related to and I am grateful to Ira Gessel for calling to my attention

the relevance of the pap er Regard the vertices of G as commuting

indeterminates and dene for each integer i a p olynomial

X Y

e v

S v S

where S ranges over all ielement stable subsets of the vertex set V of G In

G G

particular e We regard e as a Ganalogue of the ith elementary

symmetric function e Indeed when G has no edges then e e v v

i i d

is not in general a symmetric where V fv v g Note however that e

function of the vertices of G

Let denote the ring of symmetric functions over Z in the variables

x x and let ZV denote the p olynomial ring over Z in the vertices of

G Dene a ring homomorphism ZV by setting e e

G G i

Since the e s for i are algebraically indep endent and generate

G G

is welldened For f we write f f f v and

G G

regard f as a Ganalogue of f

Closely related to Ganalogues of symmetric functions are certain graphs

constructed from G If V N then dene G to b e the graph obtained

from G by replacing each vertex v of G by a clique complete subgraph

K of size v and placing edges connecting every vertex of K to

v v

every vertex of K if uv is an edge of G If v then we are simply

deleting the vertex v The graphs G are usually called clan graphs and

their chromatic p olynomials have b een investigated in

Note Given V N a multicoloring of G of type is an assignment

of v distinct colors to each vertex v The multicoloring is proper if all

colors assigned to adjacent vertices are dierent If v for all v then

a multicoloring is just an ordinary coloring We can dene a symmetric

function X in exact analogy to X by

a a

2 1

x x X

where the sum ranges over all multicolorings of G of type and where a is

the number of vertices for which one of its colors is i It is evident that

X v X

v V

Thus the theory of multicolorings of G is equivalent to the theory of ordi

nary colorings of the G s and it is basically a matter of taste which one is

preferred Gasharov deals with multicolorings His result that X is

sp ositive for incomparability graphs of free p osets actually follows

from the case of ordinary colorings since if G is the incomparability graph of

a free p oset then so is each G

The following result p ointed out to me by Ira Gessel shows the con

nection b etween X and the Ganalogues e If V N then we write

v v Also write v f v for the co ecient of v in the p olyno

v V

mial or p ower series f v

Prop osition Let

T x v m xe v

summed over al l partitions Then

x v v T x v X

v V

Pro of To obtain a monomial v in the expansion of e v we must

choose stable sets S S of vertices such that S and such that

i i

each vertex v app ears in exactly v of the S s Hence

X X

v T x v m x

S S

1 2

where S S have the meaning just explained The co ecient of a mono

1 2

mial x x x in v T x v is therefore equal to the number of se

quences S S of stable sets of vertices such that S for all i and

i i

each vertex v app ears in exactly v of the S s If we color the vertices in S

i i

with the color i then we have exactly a multicoloring of G of type Hence

v T x v X x Comparing with equation completes the pro of 2

Corollary a The fol lowing three conditions are equivalent

i G is spositive for al l V N

N V for al l partitions ii s

iii Every minor of the innite Toeplitz matrix e where we set

j i

e if k has nonnegative coecients

b G is epositive for al l V N if and only if m N V for al l

partitions

Pro of a Consider the Cauchy pro duct

C x y x y

i j

xs y s

When we apply the homomorphism acting on the y variables only we

obtain

xs v s T x v

By Prop osition we have

Y X

xv s v x v s X

v V

From this the equivalence of i and ii is immediate

By the dual form of the JacobiTrudi identity every minor of

the matrix e is a skew Schur function s for suitable partitions

j i ij

G G

of a and Hence every minor of the matrix e is a Ganalogue s

j i

o ccurs as a minor Now skew Schur function Moreover every p ossible s

every skew Schur function is sp ositive and so every minor

G G

of the matrix e has nonnegative co ecients if and only if every s

j i

has nonnegative co ecients Hence ii and iii are equivalent

b This is proved exactly as the equivalence of i and ii in a using

the identity

C x y m xe y 2

We next consider the Ganalogue of the p ower sum symmetric functions

We rst note that it follows from the well known identity equivalent to

t t

log e t e t e t p t p p

that

t t

G G G G G G

log e t e t e t p t p p

Hence Theorem b elow can b e interpreted as a statement ab out the co ef

cients in the expansion of the lefthand side of

N V Theorem For al l graphs G and al l partitions we have p

is a polynomial with nonnegative integral coecients ie p

G G G G

First pro of It suces to prove the result for p since p p p

1 2

A combinatorial interpretation of the co ecients of p is an immediate conse

quence of known results in the CartierFoata theory of commutation monoids

sp ecically the result Prop in Viennots development of this theory

in terms of heaps of pieces Using the terminology of Def dene P

to b e the set of vertices of G and dene a binary relation C on P by uC v if

1 2

uv is an edge of G or u v Then the co ecient of v v v in p

where i is equal to the number of nonisomorphic pyramids heaps

with a unique maximal piece E such that v

i i

Second pro of Using the notation of the pro of of Corollary and of

we have from that

p xp y C x y z

Hence

Y X

x X v z p xv p v

v V

It is clear that p v has integral co ecients since p is an integral linear

combination of the e s It follows from that each p v has nonnegative

in terms of the basis co ecients if and only if the expansion of each X

p has nonnegative co ecients But this was shown in Cor so

the pro of follows 2

Examination of the pro of of Cor shows in fact that the co ecient

of v in p v is given by

n j j n

v p v

n where j j the number of vertices of G and where n

j G

n of the graph denotes the co ecient of n in the chromatic p olynomial

There is an interesting application of Theorem to the f vectors of

simplicial complexes For the basic notions ab out simplicial complexes used

here see eg Let b e a simplicial complex on the vertex set V

Following Tits p we call a ag complex if every minimal set

of vertices which is not a face of has two elements For instance the

order complex of a p oset p is a ag complex If G is a graph

then the collection of stable sets of vertices called the stable set complex

or independence complex of G is a ag complex and every ag complex

arises in this way Equivalently by lo oking at the complementary graph

ag complexes are the same as clique complexes of graphs ie the collection

of all sets of vertices which form a clique Let f f denote the

i i

number of ielement faces of so f unless The vector

f f f is called the f vector of A basic problem of graph

theory is to obtain information on the p ossible f vectors of ag complexes

For instance the famous theorem of Turan eg has this form

As an immediate consequence of Theorem we have the following result

which gives some nonlinear inequalities that must b e satised by f vectors

of ag complexes

Corollary Suppose that is a ag complex with f vector f f

Let

k log f t f t f t

Then each k is a nonnegative integer

Pro of Regard as the stable set complex of a graph G Set each v

G G

in Then e f while p N by Theorem 2

i i

What kind of information ab out the f vector of ag complexes is implied

by Corollary We show that it is strong enough though just barely to

establish Turans theorem for triangles rst proved by Mantel stated

b elow as Corollary Similar reasoning may b e found in where Cartier

Foata theory mentioned in our rst pro of of Theorem is used to prove

some strengthenings of Corollary The results in only use the fact

that the exp onential of the righthand side of has nonnegative co ecients

so it would b e interesting to see whether Corollary itself or the stronger

Theorem can lead to even more general results

Lemma Let a and b be positive real numbers and set

k log at bt

If each k then b a

Pro of Supp ose that the p olynomial at bt has real zeros Then the

discriminant a b is nonnegative as desired So assume that at bt

t t where C R and denotes complex conjugation

n n n

Then k where denotes the real part of a complex

number Since R it is easy to see that some p ower has negative real

part contradicting the hypothesis that k 2

Corollary Let G be a trianglefree ie no induced K graph on

d vertices without loops or multiple edges Then G has at most d edges

Pro of Let b e the clique complex of G with f vector f f By

hypothesis f f so the pro of follows from Corollary and

Lemma 2

Note that Lemma fails if we only assume that some nite number

k k k of the k s are nonnegative no matter how large N is For we

N i

can choose to have a large real part and an imaginary part very close to

zero in which case will b e p ositive unless n is large Thus Corollary

is just sucient to imply Turans theorem for triangles It is therefore no

surprise that Corollary fails to imply Turans theorem for K free graphs

For instance a graph with vertices and no K can contain at most edges

yet all co ecients of log t t t are p ositive

As a nal application of Ganalogues of symmetric functions we give

a connection with the theory of total p ositivity We will use the following

fundamental result of Aissen Schoenberg and Whitney characterizing

when a p olynomial has negative real zeros

Lemma AissenSchoenbergWhitney Let a a a R

with some a The fol lowing two conditions are equivalent

i Every zero of the polynomial a a t a t is a nonpositive real

number

ii Every minor of the innite Toeplitz matrix a where we set

j i ij

a if k or k d is nonnegative

Theorem Let G be a graph with vertex set V fv v g

such that for every V N the graph G is spositive Equivalently by

Corollary a s N V for al l partitions Let c be the number of

ielement stable sets of vertices of G Then al l the zeros of the polynomial

C t c t called the stable set p olynomial of G are real

G i

Pro of By Corollary a every minor of the Toeplitz matrix Av

e has nonnegative co ecients If we set each v in A then we

ij i

j i

obtain the matrix A c Hence every minor of A

j i ij

is nonnegative so by Lemma every zero of the p olynomial c t is real

and nonp ositive 2

Combining Theorems and yields the following result

Corollary Let P be a free poset Let c be the number of

ielement chains of P Then every zero of the polynomial c t is real 2

For a general discussion of the use of Lemma to show that combinato

rially dened p olynomials have real zeros see For additional information

on stable set p olynomials see and the references given there

A sp ecial case of Corollary are the stable set p olynomials c t of

indierence graphs also called unit interval graphs which are the incom

parability graphs of p osets that are b oth and free see eg

p These graphs have such a simple structure that there might b e

a pro of of Corollary for them that avoids Lemma p erhaps similar to

Thm

If G is a clawfree graph then every G is also clawfree Hence an imme

diate consequence of Theorem is the following

Corollary If Conjecture is true then the stable set polynomial

of a clawfree graph has only real zeros

The conclusion to the ab ove corollary was rst suggested by Hamidoune

p It is true for line graphs a sp ecial class of clawfree graphs by

a result of Heilmann and Lieb see also Cor as mentioned

by Hamidoune

A more precise connection than Theorem b etween Schur p ositivity

and the reality of the zeros of the stable set p olynomial is given as follows

Theorem Let P t be a polynomial with real coecients satisfying

P Dene

F x P x

P i

an inhomogeneous symmetric formal power series The fol lowing three con

ditions are equivalent

i F x is spositive Equivalently every homogeneous component of

F x is spositive

ii F x is epositive

iii Al l the zeros of P t are negative real numbers

Pro of If P t t with then by we have

j j

F x m e x

where in general f f From this it is clear that iiiii

while iii is obvious since each e is sp ositive Now also from we

have

s x F x s

Arguing as in the pro of of Corollary it follows from Lemma that

F x is sp ositive if and only if each is a p ositive real number Hence

P i

iiii and the pro of follows 2

Corollary The fol lowing three conditions on a graph G with

vertex set V are equivalent

i The symmetric function

Y X

V N

is spositive

ii Y is epositive

iii Al l the zeros of the stable set polynomial C t of G are real

Pro of A simple combinatorial argument shows that

Y x C x

G G i

The pro of follows from Theorem and the fact that C t has p ositive

co ecients so every real zero is negative 2

Generalizations

There are a number of p ossible generalizations of the symmetric function

X These generalizations are largely unexplored territory We will sketch

what is known ab out three such generalizations in this section

The Tutte p olynomial

The Tutte p olynomial T x y is a p olynomial in two variables asso ciated

with a graph G or more generally any matroid It sp ecializes to the chro

matic p olynomial via the identity Unlike the chromatic p olynomial

the Tutte p olynomial do es not vanish when the graph has lo ops and is not

unaected by replacing a multiple edge by a single edge Hence we will allow

G to have lo ops and multiple edges For a go o d survey of Tutte p olynomials

see One of the formulas Prop for the Tutte p olynomial of a

graph though not the original denition is given by

t n

G m

t T t t

t n

V n

where a cG denotes the number of connected comp onents of G b G

denotes the rank of the b ond lattice L ie G V cG c ranges

over al l colorings of G with the n colors n f ng and d m

denotes the number of mono chromatic edges of number of edges of G

whose vertices are colored the same Note that if we set t in the

righthand side b ecomes n n so

G cG

n n T n

G G

Equation suggests the following symmetric function generalization

of the Tutte p olynomial

Denition Let G b e a graph on the vertex set V allowing lo ops

and multiple edges Let x x x and t b e indeterminates and dene

X x t t x

V P

where the sum is over al l colorings V P of G with p ositive integers

and where x is given by and m is as in

Note that X x X x Moreover it follows from that

G G

t n

n cG G

X t n t T t

G G

The only interesting results we know ab out X x t concern its expansion

in terms of p ower sum symmetric functions We will just state the main

result here rst observed by Timothy Chow The pro of is a straightforward

generalization of Thm the case t

Theorem We have

t p x X x t

S G

S E

where the sum ranges over al l subsets of the edges of G and where S is the

partition whose parts are the number of vertices of the connected components

of the spanning subgraph of G with edge set S In particular the coecients

of X x t when expanded in terms of power sum symmetric functions are

polynomials in t with nonnegative integer coecients

It is not dicult to compute X x t when G is the complete graph K

G d

To obtain a coloring satisfying i choose the sets B i

i i

Hence in ways Each such coloring satises m

1 2

m t X x t

Equivalently

Y X X

X x t x t

d m

i m

Now consider equation We can choose a subset S E by choosing

a partition fB B B g of V and placing a connected graph on each

blo ck B The contribution of a xed partition of type ie with blo ck

sizes to the righthand side of is C tC t where

1 2

C t c t

m mi

and where c is the number of connected simple graphs with i edges on

an melement vertex set Hence

X x t b C tC t p

1 2

where b is the number of partitions of type of a delement set The

numbers b are given explicitly by

m m

1 2

m m

where has m parts equal to i A simple application of the exp onential

formula eg xVI yields

d m

X X

u u

X x t C tp x exp

K m m

d m

One can also easily derive directly from

Directed graphs

Let D b e a directed graph allowing lo ops edges u u and bidirected edges

edges u v and v u u v but not multiple edges Recently Chung and

Graham dened a p olynomial C m n asso ciated with the directed

graph D C m n has many prop erties comparable with the Tutte p olyno

mial though it is not a true analogue of the Tutte p olynomial One of the

formulas for C m n though not the original denition is given as follows

Dene a pathcycle cover of D to b e a subset S of the edges such that every

comp onent of the spanning subgraph D of D with edge set S is a directed

path p ossibly of length zero ie a single vertex or directed cycle p ossibly

of length one ie a lo op from a vertex to itself Let c i j denote the

number of pathcycle covers with i paths and j cycles Then

C m n c i j m n

D D i

where m mm m i This formula suggests dening a

function x y which is symmetric separately in the two sets of variables

x x x and y y y as follows If S is a pathcycle cover

then dene S resp ectively S to b e the partition whose parts are

the number of vertices in the comp onents of D that are directed paths

resp ectively directed cycles Hence jj jj d the number of vertices of

D We now dene

x y m xp y

D S S

where the sum is over all pathcycle covers of D and where m denotes the

augmented monomial symmetric function as dened in x It follows

immediately that

m n

C m n

D D

The pathcycle symmetric function x y was investigated by Chow

We will state one of his more interesting results here which when

m n

sp ecialized to x and y answers a question raised by Chung and

Graham xc and which has no counterpart for the symmetric function

X x

Theorem Let D be a digraph with vertex set V and edge set

E V V Let D denote the complement of D ie the digraph with vertex

set V and edge set V V E Then

x y x y

xxy D x

where a denotes the involution acting on the x variables only b y

y y and c x x y means that we replace the x variables with

the union of the x and y variables

Hyp ergraphs

A simple graph may b e regarded as a set of vertices and twoelement subsets

of vertices What happ ens if we can take arbitrary subsets of vertices A

collection H of subsets of a vertex set V is called a hypergraph The elements

of H are still called edges From now on we will assume that every edge has

at least two elements We do not require as is sometimes done that the

union of the edges is V A proper coloring of H with p ositive integers is

a map V P such that no edge is mono chromatic This is equivalent

to assuming that no minimal edge is mono chromatic so we might as well

assume H is an antichain ie no two elements of H are comparable with

resp ect to inclusion

There is an extensive theory of hypergraph coloring eg Ch

but little of this theory is enumerative Given an antichain H of subsets of

V we can dene a symmetric function X exactly in analogy with graphs

X x x

where the sum ranges over all prop er colorings V P of H The only

results of any signicance we know at present ab out X concern its expansion

into p ower sum symmetric functions Let b e the lattice of partitions of

It may seem more natural to dene a coloring to b e prop er if every edge has all its

vertices colored dierently However a prop er coloring of H would then just b e a prop er

coloring of the ordinary graph whose edges are the twoelement subsets of edges of H so

nothing new would b e obtained

V and dene L to b e the joinsublattice of generated by all partitions

H V

with a unique nonsingleton blo ck B H including the empty join the

partition of V all of whose blo cks are singletons Thus if H is a graph

then L is just the lattice of contractions or b ond lattice of H There is

a further interpretation of the p oset actually a lattice since it is a nite

v g H joinsemilattice with L Let V fv v g If S fv

i H d i

then let H denote the subspace of K where K is a eld usually taken to

b e R or C given by

z g H fz z K z

i S d i

Then L is just the intersection lattice as dened in of the subspace

arrangement A fH S H g

H S

Theorem With H as above we have

p X

type

where type is the partition of d whose parts are the block sizes of

The pro of is exactly analogous to that of Thm Unlike the case of

graphs the sign of the integer do es not dep end only on type so we

cannot conclude that X is pp ositive as was the case for graphs Cor

If we set x in ie x x x x x

n n n

then X is just the chromatic p olynomial n of H ie the

H H

number of prop er colorings of H with n colors The p olynomial n is also

known as the characteristic polynomial of the subspace arrangement L

Our second result concerning the expansion of X in terms of p ower

sums is a generalization of Thm In fact it applies to an even

more general situation which generalizes X x in exactly the same way

that X x t dened in Denition generalizes X x Namely for any

G G

hypergraph H with vertex set V dene

X x t t x

V P

where the sum ranges over all colorings V P of H and where m is

the number of mono chromatic edges of H Thus X x X x Unlike

H H

the situation for X x the symmetric function X x t is not determined

H H

by the minimal elements of H so we should no longer assume that H is an

antichain Comparing with suggests that we dene the Tutte p olynomial

T x y of H by

t n

n cH H

X t n t T t

H H

where cH is the number of connected comp onents of H and H is the rank

of the intersection lattice of the arrangement A It might b e interesting to

investigate this hypergraph Tutte p olynomial further It is actually not

cH H

always a p olynomial so p erhaps the factor n t in needs to b e

mo died An alternative denition of the Tutte p olynomial of a hypergraph

has b een oered by Athanasiades x

Theorem extends to X x t in an obvious way We omit the pro of

which is completely analogous to that of Thm

Theorem Let H be a hypergraph with edge set E Then

X x t t p x

H S

S E

where the sum ranges over al l subsets of the edges of H and where S is the

partition whose parts are the number of vertices of the connected components

of the spanning subhypergraph of H with edge set S

As an explicit example if H has vertices a b c d and edges ac cd abc

then

X x t m t m t t m t m t m

p tp t tp t t p

Note that if we set t in then we obtain a second expansion the

rst b eing Theorem of X x in terms of p ower sums

As an interesting example of a hypergraph x k and let H

H consist of al l k element subsets of the delement set V The arrange

ment A is called a k equal arrangement and has b een extensively studied

By denition we have

X x x

summed over all colorings V P such that i k for all i

Standard prop erties of exp onential generating functions eg Cor

yield

d k

X Y

u u u

X x x u x x

H i

d k

Setting x x x x gives

n n n

d k

u u u

n u

d k

a result rst obtained in Cor and second equation on p see also

Thm iii using less combinatorial reasoning If we dene complex

numbers by

u u

then it follows from and the Cauchy formula that

X X

z p p xu X x

Hence by for xed d we have

d z p

type =

Similarly from we obtain

s x X x d s

the expansion of X x in terms of Schur functions Alternatively since

e i k i

where P if P is true and if P is false see for a discussion of

this notation it follows from the dual JacobiTrudi identity p

in is given by that the co ecient s

i j k

d det s

i j

If k then no index i j of an entry of the ab ove determinant

satises i j k It is not dicult to deduce that in this case we have

f the number of standard Young tableaux of shap e This fact s

is also easy to obtain from and the MurnaghamNakayama rule

It is also not dicult to compute the Tutte symmetric function X x t

of the hypergraph H It is an immediate consequence of the denition of

X x t that

1 X

# (i)

x X x t t

V P

Equivalently

d m

X Y X

u ux t

X x t

d m

i m

an immediate generalization of b oth and

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