Explicit marginal distributions for permutations with prescribed Robinson-Schensted shape

TL;DR

This paper derives explicit formulas for the local permutation probabilities conditioned on shape, revealing intricate patterns and asymptotic fixed point behavior for specific shapes using tableau methods.

Contribution

It provides the first explicit formulas for local permutation probabilities given shape, especially for hook, two-row, and rectangular shapes, using tableau-theoretic techniques.

Findings

Explicit formulas for $P^ ext{shape}_{ij}$ for certain shapes.

Patterns in permutation matrices resemble diffraction.

Expected fixed points relate to Wallis integrals as shape parameters grow.

Abstract

Given a permutation $\sigma$, the Robinson-Schensted correspondence determines a certain partition called the shape of $\sigma$. Famously, the shape measures the longest unions of increasing and decreasing subsequences, thus giving global information about $\sigma$. In this paper, by contrast, we ask how prescribing a shape collectively controls local behavior: namely, if $\sigma$ is a random permutation of shape $\lambda$, then what is $P^\lambda_{ij} :=$ the probability that $\sigma(i) = j$? Using tableau-theoretic methods, we derive explicit formulas for $P^\lambda_{ij}$ when $\lambda$ is a hook, two-row, or rectangular shape. We use these formulas to depict and analyze the intricate diffraction-like patterns in the matrices $(P^\lambda_{ij})$. As a surprising application, we show that for both hook and two-row shapes, as the largest part of $\lambda$ tends to infinity with the remaining parts fixed (summing to $m$), the expected proportion of fixed points in $\sigma$ approaches the Wallis integral $\int_0^{\pi/2} \sin^{2m+1} x \: dx = (2m)!! / (2m+1)!!$.