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An equivalent formulation of chromatic quasi-polynomials
Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-10-01 , DOI: 10.1016/j.disc.2020.112012
Tan Nhat Tran

Abstract Given a central integral arrangement, the reduction of the arrangement modulo a positive integer q gives rise to a subgroup arrangement in Z q l . Kamiya et al. (2008) introduced the notion of characteristic quasi-polynomial, which enumerates the cardinality of the complement of this subgroup arrangement. Chen and Wang (2012) found a similar but more general setting that replacing the integral arrangement by its restriction to a subspace of R l , and evaluating the cardinality of the q -reduced complement will also lead to a quasi-polynomial in q . On an independent study, Branden and Moci (2014) defined the so-called chromatic quasi-polynomial, and initiated the study of q -colorings on a finite list of elements in a finitely generated abelian group. The main purpose of this paper is to verify that the Chen–Wang quasi-polynomial and the Branden–Moci chromatic quasi-polynomial are equivalent in the sense that the quasi-polynomials enumerate the cardinalities of isomorphic sets. Some applications including periodicity of the intersection posets of Z q -arrangements, an answer to a problem of Chen–Wang, and computation on the characteristic polynomials of R -arrangements will also be discussed.

中文翻译:

色准多项式的等效公式

摘要 给定中心积分排列,以正整数 q 为模的排列的约简产生 Z ql 中的子群排列。神谷等人。(2008) 引入了特征拟多项式的概念,它列举了这个子群排列的补集的基数。Chen 和 Wang (2012) 发现了一个类似但更一般的设置,即通过其对 R l 的子空间的限制来代替积分排列,并且评估 q 约简补的基数也将导致 q 中的拟多项式。在一项独立研究中,Branden 和 Moci (2014) 定义了所谓的色拟多项式,并在有限生成的阿贝尔群中的有限元素列表上启动了 q 着色研究。本文的主要目的是验证 Chen-Wang 拟多项式和 Branden-Moci 色拟多项式在拟多项式枚举同构集的基数的意义上是等价的。还将讨论一些应用,包括 Z q 排列的交集的周期性,对陈-王问题的解答,以及 R 排列的特征多项式的计算。
更新日期:2020-10-01
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