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

TL;DR

This paper calculates the probability that a cell in a specific shifted staircase Young tableau contains the maximum label and demonstrates that its asymptotic distribution follows the quarter-circle law, linking combinatorics and probability.

Contribution

It explicitly determines the probability distribution of the maximal cell in shifted staircase tableaux and establishes its asymptotic behavior governed by the quarter-circle law.

Findings

Probability of maximal label in a cell explicitly computed

Asymptotic distribution follows the quarter-circle law

Connections made between tableaux, group decompositions, and sorting networks

Abstract

In this note, we explicitly compute the probability that a given cell in a random standard Young tableau of the shifted staircase shape $(2n-1, 2n-3, \ldots, 3,1)$ 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 $B_n$.