An analytic derivation of a generating function for $k$-alternating permutations

Abstract

We study the inversion enumerator of permutations whose descent set is fixed to be the set of multiples of a fixed integer $k\ge 2$. For each $n$, let ${\mathfrak{S}}_n^{(k)}$ denote the set of permutations of $\{1,\dots ,n\}$ whose descent set is exactly $\{i:k\mid i\}$, and define the polynomial $$a_n^{(k)}(q)=\sum _{\ddot{I}ƒ\in {\mathfrak{S}}_n^{(k)}}{q}^{\mathrm{inv}(\ddot{I}ƒ)}.$$ We prove that the associated $q$-exponential generating function $$F_k(t;q)=\sum _{n\ge 0}\frac{t^n}{[n{]}_q!}{a}_n^{(k)}(q),$$ where $[n{]}_q!$ denotes the $q$-factorial, admits an explicit closed form as a ratio of two $k$-periodically truncated $q$-exponential series. The proof is purely analytic and is based on a functional equation satisfied by $F_k(t;q)$, obtained via a decomposition of the $q$-exponential series into residue classes modulo $k$. Coefficient extraction yields a convolution identity involving Gaussian binomial coefficients, which uniquely determines the inversion enumerator. This provides an analytic alternative to classical inclusion--exclusion and structural combinatorial arguments for permutation classes with periodic descent constraints.