Descent c-Wilf Equivalence

Abstract

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

arXiv admin note: text overlap with arXiv:1510.04319

References in corpus (1)

• Anick-type resolutions and consecutive pattern avoidance