R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....colored blue, ii) there is a unique given row where all the boxes are colored red, and (iii) there is no column where all the boxes are colored blue. 5 The chromatic symmetric function A natural generalization of the chromatic polynomial, known as the chromatic symmetric function was defined in [18], and it is natural to ask whether we can explicitly compute these for Ferrers graphs. This would give us a set of symmetric functions other than the Schur functions that can be computed from Ferrers diagrams. Observe that unlike the Schur functions, the chromatic symmetric functions of Ferrers ....

.... 1, n , #, #) is the partition determined by the block sizes of # and # , and r i is the multiplicity of i in (#, #) Proof: Recall that a stable partition of the vertices of a graph G is a partition of the vertices such that each block is totally disconnected. Then Proposition 2. 4 in [18] states ) m (#) where the sum ranges over all stable partitions # of the graph G. The result follows by noting that in the complete bipartite graph K n,m , every block in a stable partition either lies entirely in the n vertices 0 , u n 1 or lies entirely in the m vertices ....

R. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math. 111 (1995), 166--194.

....HV and either the previous string of equalities or the one before that, respectively, hold. So in all cases ( S) 3. The (3 1) free Conjecture In order to make a connection of our work with the (3 1) free Conjecture, we rst need to introduce Stanley s chromatic symmetric function [11, 12]. Let G = V; E) be a graph with a nite set of vertices V and edges E. A proper coloring of G from a set A is a function : V A such that uv 2 E implies (u) 6= v) Now consider a countably in nite set of variables x = fx 1 ; x 2 ; g. Stanley associated with each graph a formal power ....

....set with n elements. So under this substitution, XG (1 ) PG (n) where PG (n) is the famous chromatic polynomial of Whitney [17] Also, because permuting the colors of a proper coloring keeps the coloring proper, XG (x) is in the algebra (x) of symmetric functions in x over the rationals. In [11, 12], Stanley was able to derive many interesting properties of the chromatic symmetric function XG (x) some of which generalize those of the chromatic polynomial and some of which cannot be interpreted after substitution. One natural question to ask is whether one can say anything about the ....

R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.

....study of the umbral chromatic polynomial (H; x) of a simple graph H. This viewpoint has already proven fruitful in the context of applications to algebraic topology [5] and has been amplified by related advances in algebraic combinatorics, such as Stanley s symmetric function generalization [16] of the classical chromatic polynomial (H;x) which encodes the same information as our (H; x) The results of this paper are used in [6] and [7] to carry out certain computations, and to construct combinatorial models for several Hopf algebras arising in formal group theory and algebraic ....

.... possible to state the analogue of Whitney s result (1) for (P; x) in terms of P c , although we were only able to state it in terms of P for (P; x) We conclude this section with a reference to Stanley s symmetric function XH , which generalizes the chromatic polynomial of a graph H (see [16]) We use the same notation for symmetric functions as in [16] namely p , m , and f m for the power sum, monomial, and augmented monomial symmetric functions corresponding to the partition = h1 : i of a positive integer, where the latter is defined by f m : r 1 r 2 : m : ....

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R. P. Stanley. A symmetric function generalization of the chromatic polynomial of a graph. Adv. Math., 111:166, 1995.

....polyomial of set systems of a fairly general type, which we call partitio systems. This invariant was first defined for graphs by N. Ray and C. Wright in [41] in which case it en codes the same information about the graph as R. Stanley s symmetric fuctio geeralisatio of the chromatic polyomial [51]. We propose two definitions for a colouring of a partition system, which coincide with the definition due to Wag ner [55] in the case of simplicial complexes. These new definitions of colouring enable us to generalise the product formula for the classical chromatic polyno mial of a graph, as ....

....laws, this formula may best be interpreted in the context of Hopf algebras, as in Chapter 4. Finally, the symmetric functions X(79; 2; and X(q, G; 2; are natural exten sions of Stanley s recently introduced symmetric function generalisation XH of the chromatic polynomial of a graph H (see [51])7 indeed X(Z(H) 2; XH. As Stanley points out and we discuss in detail in 52.2, the sylnmetric function XH encodes the same information as the umbral chromatic polynomial of H. By way of simple examples, of which the last two are just restating (1.7. we remark that p(v, 2; c(v, ....

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R. P. Stanley. A symmetric function generalization of the chromatic poly- nomial of a graph. Adv. Math., 111:166, 1995.

....complex I(G) of a graph G, the polynomial (S ; x) reduces to the umbral chromatic polynomial (G ; x) as documented in [15] for example) the classical chromatic polynomial of the graph may be retrieved by replacing each OE i with 1. The symmetric function X(I(G) x) was defined in [19] as another generalization of the chromatic polynomial of G; as Stanley points out, this function encodes the same information as the umbral chromatic polynomial of G. By way of simple examples, we remark that (N n ; x) c (N n ; x) x ; ae n (x) c n (x) 4.5) We also have ....

R. P. Stanley. A symmetric function generalization of the chromatic polynomial of a graph. Adv. Math., 111:166, 1995.

....of the tensor algebra TV , then the algebra of invariants of G is also a tensor algebra. Anick [1] then showed that one could remove the hypothesis about G preserving the grading. Most recently, Gebhard and Sagan [8] revived these ideas as a tool for studying Stanley s chromatic symmetric function [27, 29]. The aim of this paper is to give the first systematic study of the properties of Pi(x) We define analogues of all of the standard bases for (x) including the Schur functions. We then study the corresponding basis change equations and inner products. For the Schur functions, which are defined ....

....We should also note that Doubilet [3] also posed the problem of finding an analogue of the Schur basis indexed by set partitions in his setting. V) As mentioned in the introduction, one of the motivations for introducing noncommuting variables is to study Stanley s chromatic symmetric function [8, 27, 29]. Let G = V; E) be a graph and let PG (n) denote the chromatic polynomial of G, i.e. the number of proper colorings of G from a set with n colors. This object was first studied by Whitney [34] who proved, among other things, that it is always a polynomial in n. It has many interesting properties ....

R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.

.... have monotonically decreasing sequences, see [10, 11] proofs from other perspectives and related works appear in [8, 9, 21, 23, 24] In another direction, Stanley has developed a symmetric function generalization of the chromatic polynomial which contains the chromatic di erence sequence, see [26, 27]. 1991 Mathematics Subject Classi cation. 05C. Key words and phrases. Cayley graph, chromatic di erence sequence, circulant, graph homomorphism, independence ratio, No homomorphism lemma, partitionable graph. 1 2 KAREN L. COLLINS Let G have n vertices. The normalized chromatic di erence ....

R. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math., 111 (1995), 166-194.

....i.e. for k l 1 the polynomial m l (G; x)m k (G; x) Gamma m l Gamma1 (G; x)m k 1 (G; x) is a polynomial in the x e s (where e varies over all edges of G) with nonnegative coefficients. Remark. A generalization of this Theorem in connection with Stanley s symmetric function generalization [17, 18] of the chromatic polynomial has been recently found by Gasharov [6, Theorem 1.1] ....

R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Adv. in Math., (to appear).

....(1991) Primary 05C15; Secondary 05E05. Proposed running head: Chromatic Symmetric Function Send proofs to: Bruce E. Sagan Department of Mathematics Michigan State University East Lansing, MI 48824 1027 Tel. 517 355 8329 FAX: 517 432 1562 Email: sagan math.msu.edu Abstract In [12], Stanley associated with a graph G a symmetric function XG which reduces to G s chromatic polynomial XG (n) under a certain specialization of variables. He then proved various theorems generalizing results about XG (n) as well as new ones that cannot be interpreted on the level of the chromatic ....

....graph with verticies V = V (G) fv 1 ; v 2 ; v d g and edge set E = E(G) We permit our graphs to have loops and multiple edges. Let XG (n) be the chromatic polynomial of G, i.e. the number of proper colorings : V f1; 2; ng. Proper means that vw 2 E implies (v) 6= w) In [12, 13], R. P. Stanley introduced a symmetric function, XG , which generalizes XG (n) as follows. Let x = fx 1 ; x 2 ; g be a countably infinite set of commuting indeterminates. Now define XG = XG (x 1 ; x 2 ; X x (v 1 ) x (v d ) where the sum ranges over all proper colorings, ....

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R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.

....for this paper would be a long complicated argument for a special case of a more general theorem which has a short elegant proof . As such, we must not only present our result clearly, but also explain why doing it the hard way is interesting. First, we present the background material. In [7], Stanley introduces the chromatic symmetric function XG (x) of a (finite, simple, undirected) graph G = V; E) defined as follows. Let x : fx 1 ; x 2 ; g be independent commuting indeterminates over Q, and for a function f : V P let x f : Q v2V x f(v) Finally, XG (x) P f x f ....

.... a function f : V P let x f : Q v2V x f(v) Finally, XG (x) P f x f where the sum is over all functions f : V P such that if u v is an edge of G then f(u) 6= f(v) This generalizes the usual chromatic polynomial of G, and many interesting properties of XG (x) are developed in [7, 8]. Given a finite partial order (poset) P we use the notation ujjv to indicate that u; v 2 P are concurrent in P , meaning that u and v are either incomparable or equal; the incomparability graph Inc(P ) of P has vertex set P and edges u v whenever u 6= v and ujjv in P . A poset is (3 1) free ....

R.P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.

....(1991) Primary 05C15; Secondary 05E05. Proposed running head: Chromatic Symmetric Function Send proofs to: Bruce E. Sagan Department of Mathematics Michigan State University East Lansing, MI 48824 1027 Tel. 517 355 8329 FAX: 517 432 1562 Email: sagan math.msu.edu Abstract In [12], Stanley associated with a graph G a symmetric function XG which reduces to G s chromatic polynomial XG (n) under a certain specialization of variables. He then proved various theorems generalizing results about XG (n) as well as new ones that cannot be interpreted on the level of the chromatic ....

....graph with verticies V = V (G) fv 1 ; v 2 ; v d g and edge set E = E(G) We permit our graphs to have loops and multiple edges. Let XG (n) be the chromatic polynomial of G, i.e. the number of proper colorings : V f1; 2; ng. Proper means that vw 2 E implies (v) 6= w) In [12, 13], R. P. Stanley introduced a symmetric function, XG , which generalizes XG (n) as follows. Let x = fx 1 ; x 2 ; g be a countably infinite set of commuting indeterminates. Now define XG = XG (x 1 ; x 2 ; X x (v 1 ) x (v d ) where the sum ranges over all proper colorings, ....

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R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.

....Let v 0 be any vertex of G. Then jA(G; v 0 )j = ja 1 j: 1) Originally this theorem was proved using the theory of hyperplane arrangements. The purpose of this paper is to give three other proofs using different techniques. In the next section we will give a purely inductive proof. Stanley [9] indicated that such a proof exists and we provide the details. 1 In the paper just cited, Stanley introduced a symmetric function analog of the chromatic polynomial and showed that it counts the number of acyclic orientations of G with j sinks, 1 j d. Note that this is not quite the same as ....

....this case we can prove inductively from Theorem 2.1 that ja 1 j = 0 and from Lemma 2.3 we see that jA(G; v 0 )j = 0 as well. Thus the boundary conditions match and we are done. 3 Chromatic Symmetric Functions Using his symmetric function generalization, XG , of the chromatic polynomial, Stanley [9] proved a result related to, but not quite implying, the one of Greene and Zaslavsky. See Theorem 3.7 at the end of this section. In [3] we introduced an analogue of XG using noncommutative variables. This allows us to use deletioncontraction techniques on symmetric functions to prove a ....

[Article contains additional citation context not shown here]

R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.

....of the umbral chromatic polynomial OE (H; x) of a simple graph H. This viewpoint has already proven fruitful in the context of applications to algebraic topology [5] and has been amplified by related advances in algebraic combinatorics, such as Stanley s symmetric function generalization [16] of the classical chromatic polynomial (H;x) which encodes the same information as our OE (H; x) The results of this paper are used in [6] and [7] to carry out certain computations, and to construct combinatorial models for several Hopf algebras arising in formal group theory and algebraic ....

.... to state the analogue of Whitney s result (1) for (P; x) in terms of P c , although we were only able to state it in terms of P for OE (P; x) We conclude this section with a reference to Stanley s symmetric function XH , which generalizes the chromatic polynomial of a graph H (see [16]) We use the same notation for symmetric functions as in [16] namely p , m , and f m for the power sum, monomial, and augmented monomial symmetric functions corresponding to the partition = h1 r 1 2 r 2 : i of a positive integer, where the latter is defined by f m : r 1 r 2 : ....

[Article contains additional citation context not shown here]

R. P. Stanley. A symmetric function generalization of the chromatic polynomial of a graph. Adv. Math., 111:166, 1995.

....I(G) of a graph G, the polynomial OE (S ; x) reduces to the umbral chromatic polynomial OE (G ; x) as documented in [15] for example) the classical chromatic polynomial of the graph may be retrieved by replacing each OE i with 1. The symmetric function X(I(G) x) was defined in [19] as another generalization of the chromatic polynomial of G; as Stanley points out, this function encodes the same information as the umbral chromatic polynomial of G. By way of simple examples, we remark that ae OE (N n ; x) c OE (N n ; x) x n ; ae OE (K n ; x) B OE n (x) c ....

R. P. Stanley. A symmetric function generalization of the chromatic polynomial of a graph. Adv. Math., 111:166, 1995.

....for example, if is a set partition then the expression p is to be understood as an abbreviation for p type( We will use to denote the involution that sends s to s 0 . If d is a positive integer, we use the notation [d] for the set f1; 2; dg. Following Gessel [7] and Stanley [12], we define a power series in the countably many variables x = fx 1 ; x 2 ; g to be quasi symmetric if the coefficients of x r1 i 1 x r2 i 2 Delta Delta Delta x rk i k and x r1 j 1 x r2 j 2 Delta Delta Delta x rk j k are equal whenever i 1 i 2 Delta Delta Delta i k ....

....of V (G) such that every block is a stable set, i.e. no two vertices in the same block are connected by an edge. Stanley s chromatic symmetric function XG is defined by XG def = X m ; where the sum is over all stable partitions of G. For motivation for the definition of XG , see [12]. Here we will just mention that an equivalent definition of XG is XG = X :V (G) N Y v2V (G) x (v) where the sum is over all proper colorings , i.e. maps : V (G) N such that (u) 6= v) whenever u is adjacent to v. One can check that XG (1 n ) is just the chromatic polynomial of ....

R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.

....a P tableau. Let G be a finite simple undirected graph with vertex set V = fv 1 ; v 2 ; v d g. A proper coloring of G is a map : V N such that (v i ) 6= v j ) whenever v i and v j are adjacent. Let fxn j n 2 Ng be a countably infinite family of independent indeterminates. Following [5], define the chromatic symmetric function XG of G to be the formal power series XG def = X x (v1 ) x (v2 ) Delta Delta Delta x (vd ) where the sum is over all proper colorings of G. It is easy to see that XG is a symmetric function, so it can be written as a (finite) linear ....

R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.

....partitions of G. We are using the notation = i(i Gamma 1) Delta Delta Delta (i Gamma k 1) and i = i(i 1) Delta Delta Delta (i k Gamma 1) In particular, G (i) is a polynomial in i, as advertised. Now let x 1 ; x 2 ; be commuting independent indeterminates. Stanley [St2] defines the chromatic symmetric function XG by XG = XG (x) m (x) where the sum is over all stable partitions of G. It is easy to see that m (1 , whence XG (1 ) G (i) so that XG is a generalization of the chromatic polynomial. Stanley proves a number of results about XG in ....

....the chromatic symmetric function XG by XG = XG (x) m (x) where the sum is over all stable partitions of G. It is easy to see that m (1 , whence XG (1 ) G (i) so that XG is a generalization of the chromatic polynomial. Stanley proves a number of results about XG in the papers [St2][St4] see also [Ga1] Ga2] We shall be investigating some other properties of XG later. 21 Now let D be a digraph. Following Chung and Graham [CG1] we say that a subset S of the edges of D is a path cycle cover of D if no two elements of S lie in the same row or column of the associated ....

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R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.

....x 2 ; of G by XG (x) 1 x 2 ; summed over all proper colorings of G, where a i ( # (i) the number of vertices of G colored i. Clearly XG (x) is a homogeneous symmetric function of degree n in the variables x 1 ; x 2 ; Its basic properties are developed in [80][88] If P is a nite poset, then let inc(P ) denote its incomparability graph, i.e. the vertices of inc(P ) are the elements of P , and uv is an edge of inc(P ) if and only if u and v are incomparable in P . The theorem of Gasharov mentioned above is the following. Theorem 4. If P is a (3 ....

R. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.

....of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 e mail: rstan math.mit.edu version of 29 June 1995 Partially supported by NSF grant #DMS 9206374. 1 Schur positivity. Let G be a finite graph with no loops (edges from a vertex to itself) or multiple edges. In [36] we defined a symmetric function XG = XG (x 1 ; x 2 ; which generalizes the chromatic polynomial G (n) of G. In this paper we will report on further work related to this symmetric function. We first review the definition of XG . We will denote by V = fv 1 ; v d g the vertex set and ....

....is immediate from the definition of XG that ) G (n) where in general for a symmetric function f , we denote by f(1 ) the substitution x 1 = x 2 = Delta Delta Delta = x n = 1, x n 1 = x n 2 = Delta Delta Delta = 0. The basic properties of the symmetric function XG are discussed in [36]. In particular, we considered the expansion of XG in terms of the four bases m (the monomial symmetric functions) p (the power sum symmetric functions) s (the Schur functions) and e (the elementary symmetric functions) We are assuming a basic knowledge of symmetric functions such as may ....

[Article contains additional citation context not shown here]

R. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.

....of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 e mail: rstan math.mit.edu version of 5 December 1994 Partially supported by NSF grant #DMS 9206374. 1 Schur positivity. Let G be a finite graph with no loops (edges from a vertex to itself) or multiple edges. In [32] we defined a symmetric function XG = XG (x 1 ; x 2 ; which generalizes the chromatic polynomial G (n) of G. In this paper we will report on further work related to this symmetric function. We first review the definition of XG . We will denote by V = fv 1 ; v d g the vertex set and ....

....is immediate from the definition of XG that ) G (n) where in general for a symmetric function f , we denote by f(1 ) the substitution x 1 = x 2 = Delta Delta Delta = x n = 1, x n 1 = x n 2 = Delta Delta Delta = 0. The basic properties of the symmetric function XG are discussed in [32]. In particular, we considered the expansion of XG in terms of the four bases m (the monomial symmetric functions) p (the power sum symmetric functions) s (the Schur functions) and e (the elementary symmetric functions) We are assuming a basic knowledge of symmetric functions such as may ....

[Article contains additional citation context not shown here]

R. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math., to appear.

....X x a1 ( 1 x a2 ( 2 Delta Delta Delta ; summed over all proper colorings of G, where a i ( # Gamma1 (i) the number of vertices of G colored i. Clearly XG (x) is a homogeneous symmetric function of degree n in the variables x 1 ; x 2 ; Its basic properties are developed in [75][83] If P is a finite poset, then let inc(P ) denote its incomparability graph, i.e. the vertices of inc(P ) are the elements of P , and uv is an edge of inc(P ) if and only if u and v are incomparable in P . The theorem of Gasharov mentioned above is the following. Theorem 4. If P is a (3 ....

R. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.

....of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 e mail: rstan math.mit.edu version of 29 June 1995 1 Partially supported by NSF grant #DMS 9206374. 1 Schur positivity. Let G be a finite graph with no loops (edges from a vertex to itself) or multiple edges. In [36] we defined a symmetric function XG = XG (x 1 ; x 2 ; which generalizes the chromatic polynomial G (n) of G. In this paper we will report on further work related to this symmetric function. We first review the definition of XG . We will denote by V = fv 1 ; v d g the vertex set ....

....from the definition of XG that XG (1 n ) G (n) where in general for a symmetric function f , we denote by f(1 n ) the substitution x 1 = x 2 = Delta Delta Delta = x n = 1, x n 1 = x n 2 = Delta Delta Delta = 0. The basic properties of the symmetric function XG are discussed in [36]. In particular, we considered the expansion of XG in terms of the four bases m (the monomial symmetric functions) p (the power sum symmetric functions) s (the Schur functions) and e (the elementary symmetric functions) We are assuming a basic knowledge of symmetric functions such as ....

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R. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.

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R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166--194.

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