论文标题
关于wilf等效性的改进
On Refinements of Wilf-Equivalence for Involutions
论文作者
论文摘要
令$ \ MATHCAL {S} _n（π）$（$ \ Mathcal {i} _n（π）$和$ \ Mathcal {ai} _n（π）$）表示置于$ n $的长度$ n $的$ n $ $ n $ $ $ $π$π的$ n $的置于$ n $的订单（resp。互动和交替的参与）。对于$ k，m \ geq 1 $，backelin-west-xin证明了$ | \ mathcal {s} _n（12 \ cdotskτ）| = | \ nathcal {s} _n（k \ cdots21τ） $ k+1，k+2，\ ldots，k+m $。结果已扩展到Bousquet-Mélou和Steingrímsson的介绍，并延伸到第一作者的交替排列。在本文中，我们将建立一个峰值集，以通过$ \ Mathcal {i} _n（123τ）$和$ \ Mathcal {i} _n（321τ）$之间的横向，匹配，振荡，tableaux和成对的非交叉Dyck路径作为IntermedMedimedMedimed结构。我们的结果是对Bousquet-Mélou和Steingrímsson的结果进行了完善，因为$ k = 3 $。作为一个应用程序，我们将$ | \ Mathcal {ai} _n（123τ）| = | \ Mathcal {ai} _n（321τ）| $，确认了最近对Barnabei-Bonetti-Castronuovo-Silimbani的猜想。此外，还证明了避免避免交替互动的Barnabei-Bonetti-Castronuovo-Silimbani提出的一些猜想的平等性。
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.
