A specific geometric formula from the 1600s used to calculate pi has been found hidden inside the way we sort random lists of numbers.
Random permutations are lists of items in a random order, and they have shapes that mathematicians use to categorize them. When these lists take on a specific hook or two-row shape, the number of items that stay in their original spot follows a very specific pattern. As the lists get longer, this pattern perfectly matches the Wallis integral, a classic formula from the dawn of calculus. This reveals a deep and unexpected link between modern combinatorics and ancient geometry. It shows that even the most abstract sorting systems are governed by the same constants that define circles and triangles.
Explicit marginal distributions for permutations with prescribed Robinson-Schensted shape
arXiv · 2605.00378
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 $\sigma(i) = j$? O