Explicit marginal distributions for permutations with prescribed Robinson-Schensted shape

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 $\int_0^{\pi/2} \sin^{2m+1} x \: dx = (2m)!! / (2m+1)!!$.