Maximal Cells in Shifted Staircase Tableaux and a Quarter-Circle Law

2025/11/12 by Chihiro Kubota, Kubota, Chihiro, Taizo Sadahiro +2
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2511.09387

openalex publication_date 2025/11/12 · openalex created_date 2025/11/14 · openalex updated_date 2026/07/31

Abstract

In this note, we explicitly compute the probability that a given cell in a random standard Young tableau of the shifted staircase shape (2n-1, 2n-3, …, 3,1) contains the maximal label. We also show that the asymptotic distribution of the cell containing the maximal label is governed by the quarter-circle law. The bijection between the tableaux and thereduced decompositions of the longest element of the group B_n of the signed permutations yields the probability distribution of the first (and any) letter of the random reduced decompositions. We also show the results of some computational experiments on the random sorting networks of B_n.

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