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arXiv · 2212.01800

On Refinements of Wilf-Equivalence for Involutions

Sherry H. F. Yan·Lintong Wang·Robin D. P. Zhou
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Abstract

Let $\mathcal{S}_n(π)$ (resp. $\mathcal{I}_n(π)$ and $\mathcal{AI}_n(π)$) denote the set of permutations (resp. involutions and alternating involutions) of length $n$ which avoid the permutation pattern $π$. For $k,m\geq 1$, Backelin-West-Xin proved that $|\mathcal{S}_n(12\cdots kτ)|= |\mathcal{S}_n(k\cdots 21τ)|$ by establishing a bijection between these two sets, where $τ= τ_1τ_2\cdots τ_m$ 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 $\mathcal{I}_n(123τ)$ and $\mathcal{I}_n(321τ)$ 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 $|\mathcal{AI}_n(123τ)| = |\mathcal{AI}_n(321τ)|$, confirming a recent conjecture of Barnabei-Bonetti-Castronuovo-Silimbani. Furthmore, some conjectured equalities posed by Barnabei-Bonetti-Castronuovo-Silimbani concerning pattern avoiding alternating involutions are also proved.

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BibTeXRIS

Sherry H. F. Yan, Lintong Wang, Robin D. P. Zhou. 2022-12-04. On Refinements of Wilf-Equivalence for Involutions. https://arxiv.org/abs/2212.01800

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