Stanley's Conjecture Research Articles

A rooted variant of Stanley's chromatic symmetric function

Richard Stanley defined the chromatic symmetric function XG of a graph G and asked whether there are non-isomorphic trees T and U with XT=XU. We study variants of the chromatic symmetric function for rooted graphs, where we require the root vertex to either use or avoid a specified color. We present combinatorial identities and recursions satisfied by these rooted chromatic polynomials, explain their relation to pointed chromatic functions and rooted U-polynomials, and prove three main theorems. First, for all non-empty connected graphs G, Stanley's polynomial XG(x1,…,xN) is irreducible in Q[x1,…,xN] for all large enough N. The same result holds for our rooted variant where the root node must avoid a specified color. We prove irreducibility by a novel combinatorial application of Eisenstein's Criterion. Second, we prove the rooted version of Stanley's Conjecture: two rooted trees are isomorphic as rooted graphs if and only if their rooted chromatic polynomials are equal. In fact, we prove that a one-variable specialization of the rooted chromatic polynomial (obtained by setting x0=x1=q, x2=x3=1, and xn=0 for n>3) already distinguishes rooted trees. Third, we answer a question of Pawlowski by providing a combinatorial interpretation of the monomial expansion of pointed chromatic functions.

Some Combinatorial Properties of Skew Jack Symmetric Functions

Motivated by Stanley's conjecture on the multiplication of Jack symmetric functions, we prove a couple of identities showing that skew Jack symmetric functions are semi-invariant up to translation and rotation of a $\pi$ angle of the skew diagram. It follows that, in some special cases, the coefficients of the skew Jack symmetric functions with respect to the basis of the monomial symmetric functions are polynomials with nonnegative integer coefficients.

Resolving Stanley's conjecture on $k$-fold acyclic complexes

In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank \(1\) boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a version of the conjecture for boolean trees and show that the original conjecture holds when this notion of acyclicity is as high as possible.Mathematics Subject Classifications: 05E45, 55U10

Biconed Graphs, Weighted Forests, and $h$-Vectors of Matroid Complexes


 
 
 A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified 'coning vertices', such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs. We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of '$2$-weighted forests' of the underlying graph. This generalizes constructions of Kook and Lee who studied the Möbius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially $2$-weighted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanley's conjecture for this class of matroids. We also discuss how our constructions relate to a combinatorial strengthening of Stanley's Conjecture (due to Klee and Samper) for this class of matroids.
 
 

Reflexive polytopes arising from bipartite graphs with gamma -positivity associated to interior polynomials

In this paper, we introduce polytopes {mathscr {B}}_G arising from root systems B_n and finite graphs G, and study their combinatorial and algebraic properties. In particular, it is shown that {mathscr {B}}_G is reflexive if and only if G is bipartite. Moreover, in the case, {mathscr {B}}_G has a regular unimodular triangulation. This implies that the h^*-polynomial of {mathscr {B}}_G is palindromic and unimodal when G is bipartite. Furthermore, we discuss stronger properties, namely the gamma -positivity and the real-rootedness of the h^*-polynomials. In fact, if G is bipartite, then the h^*-polynomial of {mathscr {B}}_G is gamma -positive and its gamma -polynomial is given by an interior polynomial (a version of the Tutte polynomial for a hypergraph). The h^*-polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted. From a counterexample to Neggers–Stanley conjecture, we construct a bipartite graph G whose h^*-polynomial is not real-rooted but gamma -positive, and coincides with the h-polynomial of a flag triangulation of a sphere.

Pure $\mathcal{O}$-Sequences arising from $2$-Dimensional PS Ear-Decomposable Simplicial Complexes

We show that the $h$-vector of a $2$-dimensional PS ear-decomposable simplicial complex is a pure $\mathcal{O}$-sequence. This provides a strengthening of Stanley's conjecture for matroid $h$-vectors in rank $3$. Our approach modifies the approach of combinatorial shifting for arbitrary simplicial complexes to the setting of $2$-dimensional PS ear-decomposable complexes, which allows us to greedily construct pure multicomplex corresponding to each PS ear-decomposable complex.

On the Hilbert function of Gorenstein algebras of socle degree four

The first example of a non unimodal Gorenstein h-vector was given by Stanley, it is (1,13,12,13,1). In [14] the authors showed that Stanley's example is optimal, i.e., the vector (1,12,11,12,1) is not a Gorenstein h-vector. Our main result is a generalization of this result. We also give a simple proof of Stanley's conjecture on the asymptotic behavior of the Hilbert function in socle degree four. We present a conjecture about the asymptotic behavior of the Hilbert function for those algebras that are presented by quadrics.

Isomorphism of weighted trees and Stanley's isomorphism conjecture for caterpillars

This paper contributes to a programme initiated by the first author: "How much information about a graph is revealed in its Potts partition function?“ We show that the W -polynomial distinguishes non-isomorphic weighted trees of a good family. The framework developed to do so also allows us to show that the W -polynomial distinguishes non-isomorphic caterpillars. This establishes Stanley's conjecture for caterpillars, an extensively studied problem.

$k$-shellable simplicial complexes and graphs

In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying an expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture.Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrillón-Cruz, Cruz-Estrada and Van Tuyl-Villareal.

Internally Perfect Matroids

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 restriction on deletions, every minor of an internally perfect ordered matroid is internally perfect.

Betti Posets and the Stanley Depth

Let S be a polynomial ring and let $$I \subseteq S$$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of I determines the Stanley projective dimension of S / I or I. Our main result is that this conjecture implies the Stanley conjecture for I, and it also implies that $${{\mathrm{sdepth}}}S/I \ge {{\mathrm{depth}}}S/I - 1.$$ Recently, Duval et al. (A non-partitionable Cohen–Macaulay simplicial complex, arXiv:1504.04279 , 2015) found a counterexample to the Stanley conjecture, and their counterexample satisfies $${{\mathrm{sdepth}}}S/I = {{\mathrm{depth}}}S/I - 1$$ . So if our conjecture is true, then the conclusion is best possible.

LCM Lattices and Stanley Depth: A First Computational Approach

Let be a field, and let be the polynomial ring. Let I be a monomial ideal of S with up to five generators. In this article, we present a computational experiment that allows us to prove that . This shows that the Stanley conjecture is true for S/I and I if I can be generated by at most five monomials. The result also provides additional computational evidence for a conjecture of Herzog.

The behavior of Stanley depth under polarization

Let K be a field, R=K[X1,…,Xn] be the polynomial ring and J⊊I be two monomial ideals in R. In this paper we show thatsdepthI/J−depthI/J=sdepthIp/Jp−depthIp/Jp, where sdepthI/J denotes the Stanley depth and Ip denotes the polarization. This solves a conjecture by Herzog [9] and reduces the famous Stanley conjecture (for modules of the form I/J) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form R/I and the well-known combinatorial conjecture that every Cohen–Macaulay simplicial complex is partitionable are equivalent.

On Stanley Depths of Certain Monomial Factor Algebras

AbstractLet S = K[x1 , . . . , xn] be the polynomial ring in n-variables over a ûeld K and I a monomial ideal of S. According to one standard primary decomposition of I, we get a Stanley decomposition of the monomial factor algebra S/I. Using this Stanley decomposition, one can estimate the Stanley depth of S/I. It is proved that sdepthS(S/I) ≤ sizeS(I). When I is squarefree and bigsizeS(I) ≤ 2, the Stanley conjecture holds for S/I, i.e., sdepthS(S/I) ≥ depthS(S/I).

Lexicographic shellability, matroids, and pure order ideals

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 new approach to proving 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 matroids of rank d.

Stanley depth and simplicial spanning trees

We show that for proving the Stanley conjecture, it is sufficient to consider a very special class of monomial ideals. These ideals (or rather their lcm lattices) are in bijection with the simplicial spanning trees of skeletons of a simplex. We apply this result to verify the Stanley conjecture for quotients of monomial ideals with up to six generators. For seven generators, we obtain a partial result.

Two lower bounds for the Stanley depth of monomial ideals

Let be two monomial ideals of the polynomial ring . In this paper, we provide two lower bounds for the Stanley depth of . On the one hand, we introduce the notion of lcm number of , denoted by , and prove that the inequality holds. On the other hand, we show that , where denotes the order dimension of the lcm lattice of . We show that I and satisfy Stanley's conjecture, if either the lcm number of I or the order dimension of the lcm lattice of I is small enough. Among other results, we also prove that the Stanley–Reisner ideal of a vertex decomposable simplicial complex satisfies Stanley's conjecture.

Stanley conjecture on monomial ideals of mixed products

It is proved that the Stanley conjecture holds for monomial ideals of mixed products, i.e., if $I$ is an ideal of mixed products in a polynomial ring $S$ over a field, then ${\rm sdepth}_S(I) \geq {\rm depth}_S(I)$ and ${\rm sdepth}_S(S/I) \geq {\rm depth}_S(S/I)$.

Lexsegment Ideals Are Sequentially Cohen-Macaulay

The associated primes of an arbitrary lexsegment ideal I ⊆ S=K[x1,…,xn] are determined. As application it is shown that S/I is a pretty clean module, therefore S/I is sequentially Cohen-Macaulay and satisfies Stanley's conjecture.

Almost complete intersections and Stanley's conjecture

Let K be a field and I a monomial ideal of the polynomial ring S = K[x1, ..., xn]. We show that if either: 1) I is almost complete intersection, 2) I can be generated by less than four monomials; or 3) I is the Stanley-Reisner ideal of a locally complete intersection simplicial complex on [n], then Stanley's conjecture holds for S/I.