Gessel polynomials, rooks, and extended Linial arrangements

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:163

Article

Tewari, Vasu¹ 

[1] Univ Penn, Dept Math, Philadelphia, PA 19104 USA

关键词: Characteristic polynomial;    Descent;    Excedance;    Hyperplane arrangements;    Trees;    Rook placements;   

DOI  :  10.1016/j.jcta.2018.11.017

来源: Elsevier

【 摘 要 】

We study a family of polynomials associated with ascent- descent statistics on labeled rooted plane k-ary trees introduced by Gessel, from a rook-theoretic perspective. We generalize the excedance statistic on permutations to maximal nonattacking rook placements on certain rectangular boards by decomposing them into staircase boards. We then relate the number of maximal nonattacking rook placements on certain skew boards to the number of regions in extended Linial arrangements by establishing a relation between the factorial polynomial of those boards to the characteristic polynomial of extended Linial arrangements. Furthermore, we give a combinatorial interpretation of the number of bounded regions in extended Linial arrangements in terms of certain labeled rooted plane k-ary trees. Finally, using the work of Goldman-Joichi-White, we identify graphs whose chromatic polynomials equal the characteristic polynomials of extended Linial arrangements up to a straightforward normalization. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free