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arXiv · 2106.14762

The runsort permuton

Noga Alon·Colin Defant·Noah Kravitz
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Abstract

Suppose we choose a permutation $\pi$ uniformly at random from $S_n$. Let $\mathsf{runsort}(\pi)$ be the permutation obtained by sorting the ascending runs of $\pi$ into lexicographic order. Alexandersson and Nabawanda recently asked if the plot of $\mathsf{runsort}(\pi)$, when scaled to the unit square $[0,1]^2$, converges to a limit shape as $n\to\infty$. We answer their question by showing that the measures corresponding to the scaled plots of these permutations $\mathsf{runsort}(\pi)$ converge with probability $1$ to a permuton (limiting probability distribution) that we describe explicitly. In particular, the support of this permuton is $\{(x,y)\in[0,1]^2:x\leq ye^{1-y}\}$.

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Noga Alon, Colin Defant, Noah Kravitz. 2021-06-28. The runsort permuton. https://arxiv.org/abs/2106.14762

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