Counterexamples to the Neggers-Stanley conjecture

On operators on polynomials preserving real-rootedness and the Neggers-Stanley Conjecture

We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Toeplitz matrices. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for $E$-polynomials of series-parallel posets and column-strict labelled Ferrers posets.

Stanley关于二聚体覆盖猜想的反例

Counterexamples to Stanley's conjecture on dimer coverings

通过构造三角代数单位间的乘法恒等式，证明了Stanley关于二聚体覆盖生成函数分母多项式仅有单根的猜想在k=13时首次失效。

Stanley–Gasharov猜想的无穷反例族

Two infinite families of counterexamples to the Stanley--Gasharov conjecture

该研究通过结合无爪图的阶数普查数据，证明了Stanley–Gasharov猜想的最小反例，并构造出无穷族连通线图作为该猜想的无穷反例。

Ehrhart Polynomial Roots and Stanley's Non-negativity Theorem

Stanley's non-negativity theorem is at the heart of many of the results in Ehrhart theory. In this paper, we analyze the root behavior of general polynomials satisfying the conditions of Stanley's theorem and compare this to the known root behavior of Ehrhart polynomials. We provide a possible counterexample to a conjecture of the second author, M. Beck, J. De Loera, J. Pfeifle, and R. Stanley, and contribute some experimental data as well.

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.

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 \[ \operatorname{sdepth} S/I \geq \operatorname{depth} S/I - 1.\] Recently, Duval et al. found a counterexample to the Stanley conjecture, and their counterexample satisfies $\operatorname{sdepth} S/I = \operatorname{depth} S/I - 1$. So if our conjecture is true, then the conclusion is…