A Two-Corner Decomposition for Fixed Points of 123-Avoiding Permutations
Overview
A 123-avoiding permutation has no increasing subsequence of length three. For permutations with two fixed points, a decomposition into two corners proves the conjectured formula for their distance. It also counts their positions and excedances (entries larger than their positions), and gives distance distributions for involutions and half-turn symmetric permutations.
Original public questions
Birmajer–Gil–Tirrell–Weiner Conjecture A.2: fixed-point distance
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.
Original abstract (English)
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.
Mathematical review
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