Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infnitely many variables. The ordinary chromatic polynomial is a specialization of the Stanley's one. Our goal is to extend Stanley's chromatic polynomial to embedded graphs. In contrast to well-known extensions of, say, the Tutte polynomial from abstract to embedded graphs [5], we do not treat an embedded graph as an abstract graph endowed with additional information about the embedding. Instead, we consider the binary delta-matroid associated to an embedded graph and define the extended Stanley's chromatic polynomial as an invariant of delta-matroids. We show that, similarly to Stanley's symmetrized chromatic polynomial of graphs, which satisfies 4-term relations for simple graphs and determines in this way a knot invariant, extended Stanley's chromatic polynomial of binary delta-matroids we define satisfies the 4-term relations for binary delta-matroids [10] and determines, therefore, an invariant of links.
To each linear code over a finite field we associate the matroid of its parity check matrix. We show to what extent one can determine the generalized Hamming weights of the code (or defined for a matroid in general) from various sets of Betti numbers of Stanley-Reisner rings of simplicial complexes associated to the matroid.
In recent years we have worked on a project involving poset topology, various analogues of Eulerian polynomials, and a refinement of Richard Stanley's chromatic symmetric function. Here we discuss how Stanley's ideas and results have influenced and inspired our own work.
In this note we define a Stanley-Reisner ring for quasi-arithmetic matroids and more general structures. To this end, we define two types of CW complexes associated with a quasi-arithmetic matroid that generalize independence complexes of matroids. Then we use Stanley's construction of Stanley-Reisner rings for simplicial posets.
In 1977 Stanley conjectured that the $h$-vector of a matroid independence complex is a pure $O$-sequence. In this paper we use lexicographic shellability for matroids to motivate a combinatorial strengthening of Stanley's conjecture. This suggests that a pure $O$-sequence can be constructed from combinatorial data arising from the shelling. We then prove that our conjecture holds for matroids of rank at most four, settling the rank four case of Stanley's conjecture. In general, we prove that if our conjecture holds for all rank $d$ matroids on at most $2d$ elements, then it holds for all matro…
In this paper, we study positivity phenomena for the $e$-coefficients of Stanley's chromatic function of a graph. We introduce a new combinatorial object: the {\em correct} sequences of unit interval orders, and using these, in certain cases, we succeed to construct combinatorial models of the coefficients appearing in Stanley's conjecture. Our main result is the proof of positivity of the coefficients $c_{n-k,1^k}$, $c_{n-2,2}$, $c_{n-3,2,1}$ and $c_{2^k,1^{n-2k}}$ of the expansion of the chromatic symmetric function in terms of the basis of the elementary symmetric polynomials for the case o…
Stanley conjectured in 1977 that the $h$-vector of a matroid simplicial complex is a pure $O$-sequence. We give simple constructive proofs that the conjecture is true for matroids of rank less than or equal to 3, and corank 2. We used computers to verify that Stanley's conjecture holds for all matroids on at most nine elements.
A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible $h$-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid $h$-vectors are pure ${O}$-sequences. Finally, we completely characterize the Stanley-Reisner ideals of 1-dimensional matroid complexes.
Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement. Stanley proved a combinatorial reciprocity theorem for chromatic functions. This is considered as an Ehrhart-type reciprocity theorem for the graphic arrangement. We introduce the chromatic signed-symmetric function of a signed graph, an analogue of the chromatic symmetric function, by the integer points of the complements of the corresponding signed-graphic arran…
In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced the concept of Kazhdan-Lusztig polynomials for matroids. It is known that both the classical Kazhdan-Lusztig polynomials and the matroid Kazhdan-Lusztig polynomials can be considered as special cases of the Kazhdan-Lusztig-Stanley polynomials for locally finite posets. In the framework of Kazhdan-Lusztig-Stanley polynomials, we study the inverse of Kazhdan-Lusztig-Stanley functions and define the inverse Kazhdan-Lusztig polynomials for matroids. We also compute these polynomi…
This article is dedicated to the study of positivity phenomena for the chromatic symmetric function of a graph with respect to various bases of symmetric functions. We give a new proof of Gasharov's theorem on the Schur-positivity of the chromatic symmetric function of a $(3 + 1)$-free poset. We present a combinatorial interpretation of the Schur-coefficients in terms of planar networks. Compared to Gasharov's proof, it gives a clearer visual illustration of the cancellation procedures and is quite similar in spirit to the proof of monomial positivity of Schur functions via the Lindstrom-Gesse…
In Stanley's seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not $e$-positive, namely, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic symmetric functions are not $e$-positive. Moreover, one such family is additionally claw-free, thus establishing that the $e$-positivity of chromatic symmetric functions is in general not dependent on th…
Chromatic quasisymmetric functions of labeled graphs were defined by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric functions. In this extended abstract, we consider an extension of their definition from labeled graphs to directed graphs, suggested by Richard Stanley. We obtain an F-basis expansion of the chromatic quasisymmetric functions of all digraphs and a p-basis expansion for all symmetric chromatic quasisymmetric functions of digraphs, extending work of Shareshian-Wachs and Athanasiadis. We show that the chromatic quasisymmetric functions of proper circular arc d…
We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's expansion in Gessel's basis of fundamental quasisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisymmetric function and both the $q$-Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimp…
Stanley associated with a graph G a symmetric function X_G which reduces to G's chromatic polynomial under a certain specialization of variables. He then proved various theorems generalizing results about the chromatic polynomial, as well as new ones that cannot be interpreted at that level. Unfortunately, X_G does not satisfy a Deletion-Contraction Law which makes it difficult to apply induction. We introduce a symmetric function in noncommuting variables which does have such a law and specializes to X_G when the variables are allowed to commute. This permits us to further generalize some of …
The $e$-positivity conjecture and the $e$-unimodality conjecture of chromatic quasisymmetric functions are proved for some classes of natural unit interval orders. Recently, J. Shareshian and M. Wachs introduced chromatic quasisymmetric functions as a refinement of Stanley's chromatic symmetric functions and conjectured the $e$-positivity and the $e$-unimodality of these functions. The $e$-positivity of chromatic quasisymmetric functions implies the $e$-positivity of corresponding chromatic symmetric functions, and our work resolves Stanley's conjecture on chromatic symmetric functions of $(3+…
Stanley has studied a symmetric function generalization X_G of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of X_G in terms of elementary symmetric functions has nonnegative coefficients if G is a clawfree incomparability graph. Here we give a combinatorial interpretation of these coefficients by combining Gasharov's work on the conjecture with Egecioglu and Remmel's combinatorial interpretation of the inverse Kostka matrix. This gives a new proof of a partial nonnegativity result of Stanley. As an interesting b…
In 1977 Stanley proved that the $h$-vector of a matroid is an $\mathcal{O}$-sequence and conjectured that it is a pure $\mathcal{O}$-sequence. In the subsequent years the validity of this conjecture has been shown for a variety of classes of matroids, though the general case is still open. In this paper we use Las Vergnas' internal order to introduce a new class of matroids which we call internally perfect. We prove that these matroids satisfy Stanley's Conjecture and compare them to other classes of matroids for which the conjecture is known to hold. We also prove that, up to a certain restri…
The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of $X_G$ to $χ_G(λ)$, the chromatic polynomial, we also define a generalization $χ^k_G(λ)$ and show that evaluations of this polynomial for negative values generalize a theorem of Stanley relating acyclic orientations to the chromatic polynomial.
We establish a formalism for working with incidence algebras of posets with symmetries, and we develop equivariant Kazhdan-Lusztig-Stanley theory within this formalism. This gives a new way of thinking about the equivariant Kazhdan-Lusztig polynomial and equivariant Z-polynomial of a matroid.
To a vector configuration one can associate a polynomial ideal generated by powers of linear forms, known as a power ideal, which exhibits many combinatorial features of the matroid underlying the configuration. In this note we observe that certain power ideals associated to transversal matroids are, somewhat unexpectedly, monomial. Moreover, the (monomial) basis elements of the quotient ring defined by such a power ideal can be naturally identified with the lattice points of a remarkable convex polytope: a polymatroid, also known as generalized permutohedron. We dub the exponent vectors of th…
We give a new proof of Chung and Graham's ``G-descent expansion'' of the classical chromatic polynomial, as well as a special case of the quasi-symmetric function expansion of the path-cycle symmetric function Xi_D. Both proofs rely on Stanley's quasi-symmetric function expansion of the chromatic symmetric function X_G. We also show that Stanley's expansion suggests that a Robinson-Schensted algorithm for (3+1)-free posets---something that has been sought for unsuccessfully for some time---ought to ``respect descents'' in a certain precise sense.
We study the chromatic symmetric functions of graph classes related to spiders, namely generalized spider graphs (line graphs of spiders), and what we call horseshoe crab graphs. We show that no two generalized spiders have the same chromatic symmetric function, thereby extending the work of Martin, Morin and Wagner. Additionally, we establish that a subclass of generalized spiders, which we call generalized nets, has no e-positive members, providing a more general counterexample to the necessity of the claw-free condition. We use yet another class of generalized spiders to construct a counter…
We partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a long-standing conjecture of Stanley. As a byproduct of this theory we establish Stanley's conjecture in various cases, for example the case of Cohen-Macaulay type less than or equal to 3.
Stanley in his paper [Stanley, Richard P.: Acyclic orientations of graphs In: Discrete Mathematics 5 (1973), Nr. 2, S. 171-178.] provided interpretations of the chromatic polynomial when it is substituted with negative integers. Greene and Zaslavsky interpreted the coefficients of the chromatic polynomial in [Greene, Curtis ; Zaslavsky, Thomas: On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. In: Transactions of the American Mathematical Society 280 (1983), jan, Nr. 1, S. 97-97.]. We shall develop an invo…
Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside a r-dimensional convex polytope and bases of a rank r transversal matroid.
Kazhdan-Lusztig-Stanley polynomials are a combinatorial generalization of Kazhdan-Lusztig polynomials of for Coxeter groups that include g-polynomials of polytopes and Kazhdan-Lusztig polynomials of matroids. In the cases of Weyl groups, rational polytopes, and realizable matroids, one can count points over finite fields on flag varieties, toric varieties, or reciprocal planes to obtain cohomological interpretations of these polynomials. We survey these results and unite them under a single geometric framework.
A well-known conjecture of Stanley is that the h-vector of a matroid is a pure O-sequence. There have been numerous papers with partial progress on this conjecture, but it is still wide open. In particular, for graphic matroids coming from taking the spanning trees of a graph as bases, the conjecture is mostly unsolved. In graph theory, a set of vertices is called dominating if every other vertex is adjacent to some vertex inside the chosen set. Kook proved Stanley's conjecture for coned graphs, which is the class of graphs that are dominated by a single vertex. Cranford et al extended that re…
We consider a linear relation which expresses Stanley's chromatic symmetric function for a poset in terms of the chromatic symmetric functions of some closely related posets, which we call the modular law. By applying this in the context of (3+1)-free posets, we are able to reduce Stanley and Stembridge's conjecture that the chromatic symmetric functions of all (3+1)-free posets are e-positive to the case of (3+1)-and-(2+2)-free posets, also known as unit interval orders. In fact, our reduction can be pushed further to a much smaller class of posets, for which we have no satisfying characteriz…
In 1981, Stanley applied the Aleksandrov-Fenchel inequalities to prove a logarithmic concavity theorem for regular matroids. Using ideas from electrical network theory we prove a generalization of this for the wider class of matroids with the ``half-plane property''. Then we explore a nest of inequalities for weighted basis-generating polynomials that are related to these ideas. As a first result from this investigation we find that every matroid of rank three or corank three satisfies a condition only slightly weaker than the conclusion of Stanley's theorem.
We define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at $-1$ up to sign. Motivated by this link between acyclic orientations and the chromatic polynomial, we develop "acyclic orientation" analogues of theorems concerning the chromatic polynomial of Birkhoff, Whitney, and Greene-Zaslavsky. As an application, we provide a new proof for Stanley's sink theorem for chromatic symmetric functions $X_G$. This theorem gives a r…
We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of Stanley's formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by Bayer, Popescu and Sturmfels. We resolve the combinatorial problems posed in their paper by computing Mobius in…
We find families of simplicial complexes where the simplicial chromatic polynomials defined by Cooper--de Silva--Sazdanovic \cite{CdSS} are Hilbert series of Stanley--Reisner rings of auxiliary simplicial complexes. As a result, such generalized chromatic polynomials are determined by $h$-vectors of auxiliary simplicial complexes. In addition to generalizing related results on graphs and matroids, the simplicial complexes used allow us to consider problems that are not necessarily analogues of those considered for graphs. Some examples include supports of cyclotomic polynomials, log concavity …
A well-known conjecture of Stanley is that the h-vector of any matroid is a pure O-sequence. There have been numerous papers with partial progress on this conjecture, but it is still wide open. Positroids are special class of linear matroids that play a crucial role in the field of total positivity. In this short note, we prove that Stanley's conjecture holds for positroids.
Stanley defined the chromatic symmetric function of a graph, and Shareshian and Wachs introduced a refinement, namely the chromatic quasisymmetric function of a labeled graph. In this paper, we define the chromatic quasisymmetric function of a directed graph, which agrees with the Shareshian-Wachs definition in the acyclic case. We give an F-basis expansion for all digraphs in terms of a permutation statistic, which we call G-descents. We use this expansion to derive a p-positivity formula for all digraphs with symmetric chromatic quasisymmetric functions. We show that the chromatic quasisymme…
Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid $M$. Our main result is that these polynomials are determined by Betti numbers associated with graded minimal free resolutions of the Stanley-Reisner ideals of $M$ and so-called elongations of $M$. Generalizing Greene's theorem from coding theory, we show that the enumerator of a matroid is equivalent to its Tutte polynomial.
Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable iff G is chordal (rigid): this is another way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M, a maximal chain of modular flats of M canonically determines a chordal graph.
It is shown that the h-vectors of Stanley-Reisner rings of three classes of matroids are pure O-sequences. The classes are (a) matroids that are truncations of other matroids, or more generally of Cohen-Macaulay complexes, (b) matroids whose dual is (rank + 2)-partite, and (c) matroids of Cohen-Macaulay type at most five. Consequences for the computational search for a counterexample to a conjecture of Stanley are discussed.
We study Stanley's long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable artinian level monomial ideals, as often done in the past, to the study of properties of pure O-sequences. We propose a conjecture on pure O-sequences and settle it in small socle degrees. This allows us to prove Stanley's conjecture for all matroids of rank 3. At the end of the paper, using our method, we discuss a first possible approach to Stanley's conjecture…
This paper deals with the so-called Stanley conjecture, which asks whether they are non-isomorphic trees with the same symmetric function generalization of the chromatic polynomial. By establishing a correspondence between caterpillars trees and integer compositions, we prove that caterpillars in a large class (we call trees in this class proper) have the same symmetric chromatic function generalization of the chromatic polynomial if and only if they are isomorphic.
We give a proof of Stanley-Stembridge conjecture on chromatic symmetric functions for the class of all unit interval graphs with independence number 3. That is, we show that the chromatic symmetric function of the incomparability graph of a unit interval order in which the length of a chain is at most 3 is positively expanded as a linear sum of elementary symmetric functions.
A well-known open problem in graph theory asks whether Stanley's chromatic symmetric function, a generalization of the chromatic polynomial of a graph, distinguishes between any two non-isomorphic trees. Previous work has proven the conjecture for a class of trees called spiders. This paper generalizes the class of spiders to $n$-spiders, where normal spiders correspond to $n = 1$, and verifies the conjecture for $n = 2$.
