An equivalent formulation of chromatic quasi-polynomials
An equivalent formulation of chromatic quasi-polynomials
复制标题
色拟多项式的等价公式
DOI：
10.1016/j.disc.2020.112012
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发表时间：
2020
影响因子：
0.8
通讯作者：
Tan Nhat Tran
中科院分区：
文献类型：
--
作者：
Takuro Abe;Hiroaki Terao and Tan Nhat Tran;Tan Nhat Tran
给定一个中心积分排列，该排列模一个正整数q的约化产生Z q中的一个子群排列。Kamiya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数的基数。Chen和Wang（2012）发现了一个类似但更一般的设置，即通过将积分排列限制为R的子空间来替换积分排列，并且评估q-约化补的基数也会导致q中的拟多项式。在一项独立的研究中，Brändén和Moci（2014）定义了所谓的色拟多项式，并开始了对有限元素列表上的q-染色的研究。证明了Chen-Wang拟多项式与Brändén-Moci色拟多项式在拟多项式枚举同构集的基数意义下是等价的.本文还讨论了Z q-排列的交偏序集的周期性、Chen-Wang问题的解答以及R-排列的特征多项式的计算等应用。
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, Brändén 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 Brändén–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.