DECOMPOSING RANDOM PERMUTATIONS INTO ORDER-ISOMORPHIC SUBPERMUTATIONS

Abstract

Two permutations σ and π are ℓ-similar if they can be decomposed into subpermutations σ(1), . . . ,σ(ℓ) and π(1), . . . ,π(ℓ) such that σ(i) is order-isomorphic to π(i) for all i ∈[ℓ]. Recently, Dudek, Grytczuk, and Ruciński Variations on twins in permutations, Electron. J. Combin., 28 (2021), P3.19. posed the problem of determining the minimum ℓ for which two permutations chosen independently and uniformly at random are ℓ-similar. We show that two such permutations are O(n1/3 log11/6(n))-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalizes to simultaneous decompositions of multiple permutations.