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arXiv · 2202.10789

Decomposing random permutations into order-isomorphic subpermutations

Carla Groenland·Tom Johnston·Dániel Korándi·Alexander Roberts·Alex Scott·Jane Tan
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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.

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Carla Groenland, Tom Johnston, Dániel Korándi, Alexander Roberts, Alex Scott, Jane Tan. 2022-02-22. Decomposing random permutations into order-isomorphic subpermutations. https://arxiv.org/abs/2202.10789

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