Explicit marginal distributions for permutations with prescribed Robinson-Schensted shape

Abstract

Given a permutation $\ddot{I}ƒ$, the Robinson-Schensted correspondence determines a certain partition called the shape of $\ddot{I}ƒ$. Famously, the shape measures the longest unions of increasing and decreasing subsequences, thus giving global information about $\ddot{I}ƒ$. In this paper, by contrast, we ask how prescribing a shape collectively controls local behavior: namely, if $\ddot{I}ƒ$ is a random permutation of shape $\hat{I}»$, then what is $P^{\hat{I}}{»}_{ij}:=$ the probability that $\ddot{I}ƒ(i)=j$? Using tableau-theoretic methods, we derive explicit formulas for $P^{\hat{I}}{»}_{ij}$ when $\hat{I}»$ is a hook, two-row, or rectangular shape. We use these formulas to depict and analyze the intricate diffraction-like patterns in the matrices $(P^{\hat{I}}{»}_{ij})$. As a surprising application, we show that for both hook and two-row shapes, as the largest part of $\hat{I}»$ tends to infinity with the remaining parts fixed (summing to $m$), the expected proportion of fixed points in $\ddot{I}ƒ$ approaches the Wallis integral ${\int }_0^{\ddot{I}€/2}{\sin }^{2m+1}xdx=(2m)!!/(2m+1)!!$.

Minor typos corrected from Version 1