Decomposing random permutations into order-isomorphic subpermutations

TL;DR

This paper investigates how to decompose random permutations into order-isomorphic subpermutations, establishing bounds on the minimum number of such subpermutations needed with high probability.

Contribution

It provides tight bounds on the minimum number of subpermutations for random permutations to be k-similar, extending to multiple permutations.

Findings

Permutations are O(n^{1/3} log^{11/6}(n))-similar with high probability

The bounds are tight up to polylogarithmic factors

Results generalize to multiple permutations

Abstract

Two permutations $s$ and $t$ are $k$-similar if they can be decomposed into subpermutations $s^1, \ldots, s^k$ and $t^1, \ldots, t^k$ such that $s^i$ is order-isomorphic to $t^i$ for all $i$. Recently, Dudek, Grytczuk and Ruci\'nski posed the problem of determining the minimum $k$ for which two permutations chosen independently and uniformly at random are $k$-similar. We show that two such permutations are $O(n^{1/3}\log^{11/6}(n))$-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalises to simultaneous decompositions of multiple permutations.