Las Vergnas Cube Conjecture and reconstruction properties of the cube matroid

How many circuits determine an oriented matroid?

Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present general upper and lower bounds in the setting of general connected orientable matroids, leading to the study of subgraphs of the base graph and the intersection graph of circuits. We then consider the problem for uniform matroids which is closely related to the notion of (connected) covering numbers in Design Theory. Finally, we also devote special attentio…

On the Number of Circuit-cocircuit Reversal Classes of an Oriented Matroid

The first author introduced the circuit-cocircuit reversal system of an oriented matroid, and showed that when the underlying matroid is regular, the cardinalities of such system and its variations are equal to special evaluations of the Tutte polynomial (e.g., the total number of circuit-cocircuit reversal classes equals $t(M;1,1)$, the number of bases of the matroid). By relating these classes to activity classes studied by the first author and Las Vergnas, we give an alternative proof of the above results and a proof of the converse statements that these equalities fail whenever the underly…

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 restri…

Topological properties of active orders for matroid bases

Las Vergnas introduced several lattice structures on the bases of an ordered matroid M by using their external and internal activities. He also noted that when computing the Moebius function of these lattices, it was often zero, although he had no explanation for that fact. The purpose of this paper is to provide a topological reason for this phenomenon. In particular, we show that the order complex of the external lattice L of M is homotopic to the independence complex of the restriction M^*|T where M^* is the dual of M and T is the top element of L. We then compute some examples showing that…

刚性拟阵中的k重回路与锥化

$k$-fold circuits and coning in rigidity matroids

利用k重回路推导关于一般d维刚性拟阵\(\mathcal{R}_d\)的新结果，包括分析2 - 和、给出k重\(\mathcal{R}_d\) - 回路的k重回路性质的充分条件及扩展怀特利锥化引理。

A matroid associated with a phylogenetic tree

A (pseudo-)metric $D$ on a finite set $X$ is said to be a `tree metric' if there is a finite tree with leaf set $X$ and non-negative edge weights so that, for all $x,y \in X$, $D(x,y)$ is the path distance in the tree between $x$ and $y$. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is -- up to canonical isomorphism -- uniquely determined by $D$, and one does not even need all of the distances in order to fully (re…

铺垫型矩阵：定义方程及相关变分

Paving Matroids: Defining Equations and Associated Varieties

本文研究铺垫型矩阵的实现空间及其闭包，引入三种生成关联理想多项式的不同方法，并计算特定铺垫型矩阵的定义方程，分析点线配置的电路变分分解及实例应用。

格路 matroid 的 Ehrhart 正性

Ehrhart positivity for lattice path matroids

本文证明了所有格路 matroid 都具有 Ehrhart 正性，统一并推广了过去二十年中关于 matroid 的 Ehrhart 正性的多个结果。作者基于之前关于 fence 的 order polynomial 正性的工作，主要结果支持 Ferroni, Jochemko 和 Schröter (2022) 关于 positroid 的 Ehrhart 正性的猜想，并进一步证明所有 Schubert matroid 都具有 Ehrhart 正性，从而解决了 Fan 和 Li (2024) 的猜想，并支持 Monical, Tokcan 和 Yong (2019) 关于 Schubitopes 的 Ehrhart 正性猜想。

双锥图、加权森林与拟阵复形的 h-向量

Biconed graphs, weighted forests, and h-vectors of matroid complexes

本文通过双锥图的2-加权森林给出图拟阵h-向量的组合解释，并构造纯多重复形证明Stanley猜想对该类拟阵成立。

On the Gorensteinness of broken circuit complexes and Orlik--Terao ideals

It is proved that the broken circuit complex of an ordered matroid is Gorenstein if and only if it is a complete intersection. Several characterizations for a matroid that admits such an order are then given, with particular interest in the $h$-vector of broken circuit complexes of the matroid. As an application, we prove that the Orlik--Terao algebra of a hyperplane arrangement is Gorenstein if and only if it is a complete intersection. Interestingly, our result shows that the complete intersection property (and hence the Gorensteinness as well) of the Orlik--Terao algebra can be determined f…

关于阶为 $N$ 且直径为 $D$ 的强连通竞赛图中的 Moon 型定理

Moon-type theorems on circuits in strongly connected tournaments of order $N$ and diameter $D$

本文证明了强连通竞赛图中关于环数量下界猜想在 $d=n-3$ 情形下成立，推广了 Moon 界与 Las Vergnas 定理。