The runsort permuton

Suppose we choose a permutation π uniformly at random from Sn. Let runsort(π) be the permutation obtained by sorting the ascending runs of π into lexicographic order. Alexandersson and Nabawanda recently asked if the plot of runsort(π), when scaled to the unit square [0, 1], converges to a limit shape as n → ∞. We answer their question by showing that the measures corresponding to the scaled plots of these permutations runsort(π) converge with probability 1 to a permuton (limiting probability distribution) that we describe explicitly. In particular, the support of this permuton is {(x, y) ∈ [0, 1] : x ≤ ye1−y}.

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