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

A permutation is called mod-k-alternating if its entries are restricted to having the same remainder as the index, modulo some integer $k\ge 1.$ 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 $n\equiv k(\mathrm{m}\mathrm{o}\mathrm{d}2k)$, the excedance enumerating polynomials over the even and odd mod-k-alternating permutations, starting with a fixed remainder, are gamma-positive.