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The Ulam–Hammersley problem for multiset permutations

Published online by Cambridge University Press:  09 May 2024

CMAP, CNRS, École Polytechnique, 91120 Palaiseau, France. e-mail:


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

Research Article
© The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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