On refinements of wilf-equivalence for involutions

Abstract

Let ${\mathcal {S}}_n(\pi )$ (resp. ${\mathcal {I}}_n(\pi )$ and $\mathcal{A}\mathcal{I}_n(\pi )$) denote the set of permutations (resp. involutions and alternating involutions) of length n which avoid the permutation pattern $\pi$. For $k,m\ge 1$, Backelin–West–Xin proved that $|{\mathcal {S}}_n(12\cdots k\tau )|= |{\mathcal {S}}_n(k\cdots 21\tau )|$ by establishing a bijection between these two sets, where $\tau = \tau _1\tau _2\cdots \tau _m$ is an arbitrary permutation of $k+1,k+2,\ldots ,k+m$. The result has been extended to involutions by Bousquet-Mélou and Steingrímsson and to alternating permutations by the first author. In this paper, we shall establish a peak set preserving bijection between ${\mathcal {I}}_n(123\tau )$ and ${\mathcal {I}}_n(321\tau )$ via transversals, matchings, oscillating tableaux and pairs of noncrossing Dyck paths as intermediate structures. Our result is a refinement of the result of Bousquet-Mélou and Steingrímsson for the case when $k=3$. As an application, we show bijectively that $|\mathcal{A}\mathcal{I}_n(123\tau )| = |\mathcal{A}\mathcal{I}_n(321\tau )|$, confirming a recent conjecture of Barnabei–Bonetti–Castronuovo–Silimbani. Furthermore, some conjectured equalities posed by Barnabei–Bonetti–Castronuovo–Silimbani concerning pattern-avoiding alternating involutions are also proved.