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 $$ a_n^{(k)}(q)=\sum_{\sigma \in \mathfrak{S}_n^{(k)}} q^{\mathrm{inv}(\sigma)}. $$ 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.