Descent c-Wilf Equivalence

Abstract

Let $S_n$ denote the symmetric group. For any $σ\in S_n$, we let $\mathrm{des}(σ)$ denote the number of descents of $σ$, $\mathrm{inv}(σ)$ denote the number of inversions of $σ$, and $\mathrm{LRmin}(σ)$ denote the number of left-to-right minima of $σ$. For any sequence of statistics $\mathrm{stat}_1, \ldots \mathrm{stat}_k$ on permutations, we say two permutations $α$ and $β$ in $S_j$ are $(\mathrm{stat}_1, \ldots \mathrm{stat}_k)$-c-Wilf equivalent if the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $α$ equals the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $β$. We give many examples of pairs of permutations $α$ and $β$ in $S_j$ which are $\mathrm{des}$-c-Wilf equivalent, $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent, and $(\mathrm{des},\mathrm{inv},\mathrm{LRmin})$-c-Wilf equivalent. For example, we will show that if $α$ and $β$ are minimally overlapping permutations in $S_j$ which start with 1 and end with the same element and $\mathrm{des}(α) = \mathrm{des}(β)$ and $\mathrm{inv}(α) = \mathrm{inv}(β)$, then $α$ and $β$ are $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent.