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arXiv · 2407.21755

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

Hiranya Kishore Dey·Iswar Mahato
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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.

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Hiranya Kishore Dey, Iswar Mahato. 2024-07-31. Sign-balance of excedances over mod-k-alternating permutations and gamma-positivity. https://arxiv.org/abs/2407.21755

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