论文标题
编织布置的改进和两个参数大惊小怪-CATALAN数字
Refinements of the braid arrangement and two parameter Fuss-Catalan numbers
论文作者
论文摘要
$ \ mathbb {r}^n $中的超平面布置是仿射超平面的有限集合。超平面排列的计数区域是列举组合学的积极研究方向。在本文中，我们考虑$ \ Mathbb {r}^n $中的安排$ \ MATHCAL {a} _n^{（m）} $由$ \ {x_i = 0 \ mid I \ in [n] n \} $对于某些固定$ a> 1 $。事实证明，这种安排家族与$ a $ a $ a $的加泰罗尼亚延长安排密切相关。我们证明，$ \ Mathcal {a} _n^{（m）} $的区域数量是加泰罗尼亚数字的一定概括，称为两个参数fuss fuss food-catalan编号。然后，我们在这些区域和某些装饰的戴克路径之间表现出两者。我们还计算了特征多项式，并为其系数提供了组合解释。我们的大多数结果也将其推广到$ \ Mathcal {a} _n^{（m）} $的子排序，通过将它们与编织布置的变形联系起来。
A hyperplane arrangement in $\mathbb{R}^n$ is a finite collection of affine hyperplanes. Counting regions of hyperplane arrangements is an active research direction in enumerative combinatorics. In this paper, we consider the arrangement $\mathcal{A}_n^{(m)}$ in $\mathbb{R}^n$ given by $\{x_i=0 \mid i \in [n]\} \cup \{x_i=a^kx_j \mid k \in [-m,m], 1\leq i<j \leq n\}$ for some fixed $a>1$. It turns out that this family of arrangements is closely related to the well-studied extended Catalan arrangement of type $A$. We prove that the number of regions of $\mathcal{A}_n^{(m)}$ is a certain generalization of Catalan numbers called two parameter Fuss-Catalan numbers. We then exhibit a bijection between these regions and certain decorated Dyck paths. We also compute the characteristic polynomial and give a combinatorial interpretation for its coefficients. Most of our results also generalize to sub-arrangements of $\mathcal{A}_n^{(m)}$ by relating them to deformations of the braid arrangement.
