123-避免排列不动点的双角分解
A Two-Corner Decomposition for Fixed Points of 123-Avoiding Permutations
研究概述
123-避免排列不含长度为三的递增子序列。对于恰有两个不动点的这类排列，双角分解证明了猜想中的不动点距离公式，并给出不动点位置和超越数（值大于位置的条目数）的联合计数，以及对合排列和中心对称排列的距离分布。
原始公开问题
Birmajer–Gil–Tirrell–Weiner 猜想 A.2：不动点距离
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell and Michael D. Weiner: Pattern-avoiding stabilized-interval-free permutations — Appendix A, Conjecture A.2: the distance between the two fixed points of a 123-avoiding permutation.
原文摘要（英文）
We enumerate 123-avoiding permutations with two fixed points by their distance, their positions, and the number of excedances (entries larger than their positions). The main construction separates a permutation into two corners with decreasing boundary conditions. The count for each corner is an explicit Catalan factor that depends on its interior size and the sum of its two boundary lengths. Combining the corners gives a positive convolution; a factorial rearrangement proves the distance formula conjectured by Birmajer, Gil, Tirrell and Weiner. Keeping the placement data yields a product formula for the joint distribution of position, distance and excedances, and determines its position support. The same construction gives distance distributions for involutions (permutations equal to their inverses) and centrosymmetric permutations (those whose plots are unchanged by a half-turn), together with bivariate generating functions and exact distance moments for involutions.
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