Related papers: $M$-Shellability of Discrete Polymatroids

We prove that if $ M $ and $ N $ are finitary matroids on a common countable edge set $ E $ then they admit a common independent set $I $ such that there is a bipartition $ E=E_{M}\cup E_{N} $ for which $ I\cap E_M $ spans $ E_M $ in $ M $…

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…

We prove that for all $d \geq 1$ a shellable $d$-dimensional simplicial complex with at most $d+3$ vertices is extendably shellable. The proof involves considering the structure of `exposed' edges in chordal graphs as well as a connection…

In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients $I/J$ of monomial ideals $J\subset I$, both invariants behave monotonic with…

We say that a pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is \emph{shelling completable} if $\Delta$ can be realized as the initial sequence of some shelling of $\Delta_{n-1}^{(d)}$, the $d$-skeleton of the…

Shelling orders are a ubiquitous tool used to understand invariants of cell complexes. Significant effort has been made to develop techniques to decide when a given complex is shellable. However, empirical evidence shows that some shelling…

Let $I\subset K[x_1,\ldots,x_n]$ be a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is…

A long-standing conjecture of Stanley states that every Cohen-Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our…

We study a natural stratification of certain affine slices of univariate hyperbolic polynomials. We look into which posets of strata can be realized and show that the dual of the poset of strata is a shellable simplicial complex and in…

Building on a recent joint paper with Sturmfels, here we argue that the combinatorics of matroids is intimately related to the geometry and topology of toric hyperkaehler varieties. We show that just like toric varieties occupy a central…

We show that an 'almost strong Lefschetz' property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the $h$-vector is unimodal, peaks in…

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…

The discrete polymatroid is a multiset analogue of the matroid. Based on the polyhedral theory on integral polymatroids developed in late 1960's and in early 1970's, in the present paper the combinatorics and algebra on discrete…

We give a criterion for modular extension of rank-4 hypermodular matroids, and prove a weakening of Kantor's conjecture for rank-4 realizable matroids. This proves the sticky matroid conjecture and Kantor's conjecture for realizable…

Shellability of a simplicial complex has many useful structural implications. In particular, it was shown by Danaraj and Klee that every shellable pseudo-manifold is a PL-sphere. The purpose of this paper is to prove the shellability of the…

Let $K$ be a field, $R=K[X_1, ..., X_n]$ be the polynomial ring and $J \subsetneq I$ two monomial ideals in $R$. In this paper we show that $\mathrm{sdepth}\ {I/J} - \mathrm{depth}\ {I/J} = \mathrm{sdepth}\ {I^p/J^p}-\mathrm{depth}\…

Let $\mathbb{K}$ be a field, and let $S=\mathbb{K}[X_1, ..., X_n]$ be the polynomial ring. Let $I$ be a monomial ideal of $S$ with up to 5 generators. In this paper, we present a computational experiment which allows us to prove that…

We show that for an arbitrarily given closed Riemannian manifold $M$ admitting a point $p \in M$ with a single cut point, every closed Riemannian manifold $N$ admitting a point $q \in N$ with a single cut point is diffeomorphic to $M$ if…

We prove that all lattice path matroids are Ehrhart positive. This unifies and generalizes numerous results on the Ehrhart positivity of matroids developed over the last two decades. We rely on our previous work on the positivity of order…

We show that all finite lattices, including non-distributive lattices, arise as stable matching lattices when all agents have path-independent choice functions. This result answers an open question of Blair~\cite{blair1988lattice}. In the…