On a refinement of Wilf-equivalence for permutations

TL;DR

This paper proves a specific case of a conjecture on the Wilf-equivalence of certain permutation patterns, using a descent set preserving bijection, and applies it to settle related conjectures.

Contribution

It confirms a conjecture on $maj$-Wilf-equivalence for certain permutation patterns and constructs a descent set preserving bijection for all $k eq 2$.

Findings

Confirmed the conjecture for $m=1$ and all $k eq 2$

Constructed a descent set preserving bijection between specific pattern-avoiding permutations

Settled a related conjecture on Wilf-equivalence for permutations with fixed descent sets

Abstract

Recently, Dokos et al. conjectured that for all $k, m\geq 1$, the patterns $ 12\ldots k(k+m+1)\ldots (k+2)(k+1) $ and $(m+1)(m+2)\ldots (k+m+1)m\ldots 21 $ are $maj$-Wilf-equivalent. In this paper, we confirm this conjecture for all $k\geq 1$ and $m=1$. In fact, we construct a descent set preserving bijection between $ 12\ldots k (k-1) $-avoiding permutations and $23\ldots k1$-avoiding permutations for all $k\geq 3$. 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.