Sign-balance of excedances over mod-k-alternating permutations and gamma-positivity

TL;DR

This paper investigates the sign-balance of excedances in mod-k-alternating permutations and derangements, establishing gamma-positivity of certain excedance polynomials under specific modular conditions.

Contribution

It introduces a novel approach using matrices and determinants to analyze sign-balance and proves gamma-positivity for excedance polynomials in mod-k-alternating permutations.

Findings

Sign-balance results for excedances in mod-k-alternating permutations

Sign-balance for excedances over mod-k-alternating derangements

Gamma-positivity of excedance polynomials under modular conditions

Abstract

A permutation is called mod-k-alternating if its entries are restricted to having the same remainder as the index, modulo some integer $k \geq 1.$ In this paper, we find the sign-balance for mod-k-alternating permutations with respect to the statistic excedance. Moreover, we study the sign-balance for excedances over mod-k-alternating derangements. The results are obtained by constructing suitable matrices and connecting their determinants with the signed excedance enumeration of mod-k-alternating permutations. As an application of the signed excedance enumeration, we prove that when $n \equiv k \pmod {2k}$, the excedance enumerating polynomials over the even and odd mod-k-alternating permutations, starting with a fixed remainder, are gamma-positive.