The Ulam-Hammersley problem for multiset permutations

TL;DR

This paper extends the classical Ulam-Hammersley problem to multiset permutations where each element appears multiple times, analyzing the asymptotic behavior of longest increasing subsequences using probabilistic methods.

Contribution

It generalizes the Ulam-Hammersley problem to multiset permutations with variable element multiplicities, providing new asymptotic results.

Findings

Asymptotic behavior characterized for multiset permutations

Connection established with particle systems for analysis

Generalization of classical problem to broader setting

Abstract

We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in {1,...,n} occurs k times, where k may depend on n. This generalizes the famous Ulam-Hammersley problem of the case k=1. The proof relies on poissonization and a connection with variants of the Hammersley-Aldous-Diaconis particle system.