123 回避順列の不動点に対する二隅分解
A Two-Corner Decomposition for Fixed Points of 123-Avoiding Permutations
研究概要
123 回避順列とは、長さ三の増加部分列を持たない順列である。不動点をちょうど二つ持つこのような順列を二つの隅に分解し、不動点間距離の予想公式を証明する。さらに不動点の位置と超過数（値が位置を上回る項の数）の同時計数、および対合と半回転対称な順列の距離分布を与える。
元の公開問題
Birmajer–Gil–Tirrell–Weiner 予想 A.2：不動点間距離
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.
原文要旨（英語）
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.
数学の検証
主要結論には公開されたカーネル検証済みの証明があり、論文との対応も確認されています。これは外部査読とは別の検証です。
検証基準