Whitney numbers of arrangements via measure concentration of intrinsic volumes

Whitney Numbers for Poset Cones

Hyperplane arrangements dissect $\mathbb{R}^n$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This paper focuses on cones within the braid arrangement, consisting of the reflecting hyperplanes $x_i=x_j$ inside $\mathbb{R}^n$ for the symmetric group, thought of as the type $A_{n-1}$ …

超平面构形的单元素扩张

One-element Extensions of Hyperplane Arrangements

本文利用诱导伴随构形分类超平面构形的单元素扩张，证明多种组合不变量在相关层上恒定且保序，并得到有限域或有理情形下特征多项式的卷积公式。

半拟阵的特征多项式及其与拟阵、超平面构形和图着色的联系

Characteristic polynomials of semimatroids and their connections to matroids, hyperplane arrangements and graph colorings

该研究探究半拟阵特征多项式的性质，推广惠特尼断链定理与罗塔-赫伦-韦尔奇猜想，给出相关卷积恒等式，并引入指派拟阵建立半拟阵、超平面构形与图着色的联系。

Stirling Numbers in Braid Matroid Kazhdan-Lusztig Polynomials

Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan-Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan-Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling …

指数排列序列中水平面的级别

Level of Faces for Exponential Sequence of Arrangements

本文提出了一种计算指数序列排列中水平面数量的生成函数，并展示了其与斯特林数的关系，同时扩展了相关多项式理论。

超平面构形的 $k$-伴随

$k$-Adjoint of Hyperplane Arrangements

本文提出超平面构形的 $k$-伴随，证明其诱导的 Grassmann 流形分解等价于拟阵分解与精细 Schubert 分解，并用于分类 $k$ 维限制及证明组合不变量的反单调性。