On a refinement of Wilf-equivalence for permutations

Abstract

Recently, Dokos et al. conjectured that for all $k,m\ge 1$, the patterns $12\dots k(k+m+1)\dots (k+2)(k+1)$ and $(m+1)(m+2)\dots (k+m+1)m\dots 21$ are $maj$-Wilf-equivalent. In this paper, we confirm this conjecture for all $k\ge 1$ and $m=1$. In fact, we construct a descent set preserving bijection between $12\dots k(k-1)$-avoiding permutations and $23\dots k1$-avoiding permutations for all $k\ge 3$. As a corollary, our bijection enables us to settle a conjecture of Gowravaram and Jagadeesan concerning the Wilf-equivalence for permutations with given descent sets.