On the longest common subsequence of independent random permutations invariant under conjugation

Mohamed Slim Kammoun (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

Bukh and Zhou conjectured that the expectation of the length of the longest common subsequence of two i.i.d random permutations of size $n$ is greater than $\sqrt{n}$. We prove in this paper that there exists a universal constant $n_1$ such that their conjecture is satisfied for any pair of i.i.d random permutations of size greater than $n_1$ with distribution invariant under conjugation. We prove also that asymptotically, this expectation is at least of order $2\sqrt{n}$ which is the asymptotic behaviour of the uniform setting. More generally, in the case where the laws of the two permutations are not necessarily the same, we gibe a lower bound for the expectation. In particular, we prove that if one of the permutations is invariant under conjugation and with a good control of the expectation of the number of its cycles, the limiting fluctuations of the length of the longest common subsequence are of Tracy-Widom type. This result holds independently of the law of the second permutation.

Domaines: Combinatoire [math.CO]; Probabilités [math.PR]

Mohamed Slim Kammoun. On the longest common subsequence of independent random permutations invariant under conjugation. The Electronic Journal of Combinatorics, 2020, 27 (4), ⟨10.37236/8669⟩. ⟨hal-02309710⟩