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.