  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 $\{(x,y)\in[0,1]^2:x\leq
ye^{1-y}\}$.
