Gessel polynomials, rooks, and extended Linial arrangements

TL;DR

This paper explores the combinatorial properties of Gessel polynomials and their connections to rook placements, Linial arrangements, and graph chromatic polynomials, revealing new relationships and interpretations in algebraic combinatorics.

Contribution

It generalizes the excedance statistic to rook placements, relates rook placements to Linial arrangements, and provides new combinatorial interpretations and graph polynomial identifications.

Findings

Established a relation between rook placements and Linial arrangements' regions.

Provided a combinatorial interpretation of bounded regions in extended Linial arrangements.

Identified graphs with chromatic polynomials matching extended Linial arrangements' characteristic polynomials.

Abstract

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 boards of staircase shape. 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 the setting of 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 upto a straightforward normalization.