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The Ulam-Hammersley problem for multiset permutations

Lucas Gerin

TL;DR

This work analyzes the longest increasing and non-decreasing subsequences in a uniform random $k$-multiset permutation of size $n$, generalizing the Ulam–Hammersley problem to variable $k_n$. It combines poissonization with variants of the Hammersley–Aldous–Diaconis process to obtain non-asymptotic bounds, followed by de-poissonization to transfer results back to the original model. The main findings are first-order asymptotics: $\mathbb{E}[\mathcal{L}_{<}(S_{k_n;n})]=2\sqrt{nk_n}-k_n+o(\sqrt{nk_n})$ and $\mathbb{E}[\mathcal{L}_{\leq}(S_{k_n;n})]=2\sqrt{nk_n}+k_n+o(\sqrt{nk_n})$, with concentration results and a Bridging argument that extends the large-$k_n$ results to intermediate scales. The work connects combinatorial LIS problems with HAD-process methods and stochastic bounds, providing a framework for understanding multiset permutation growth models and offering potential insights into KPZ-type fluctuations for generalized regimes.

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.

The Ulam-Hammersley problem for multiset permutations

TL;DR

This work analyzes the longest increasing and non-decreasing subsequences in a uniform random -multiset permutation of size , generalizing the Ulam–Hammersley problem to variable . It combines poissonization with variants of the Hammersley–Aldous–Diaconis process to obtain non-asymptotic bounds, followed by de-poissonization to transfer results back to the original model. The main findings are first-order asymptotics: and , with concentration results and a Bridging argument that extends the large- results to intermediate scales. The work connects combinatorial LIS problems with HAD-process methods and stochastic bounds, providing a framework for understanding multiset permutation growth models and offering potential insights into KPZ-type fluctuations for generalized regimes.

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.
Paper Structure (17 sections, 18 theorems, 110 equations, 6 figures)

This paper contains 17 sections, 18 theorems, 110 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Strategy of proof and organization of the paper.
  3. Beyond expectation.
  4. Comparison with previous works.
  5. Preliminaries: the case of small $k_n$
  6. Poissonization: variants of the Hammersley process
  7. Definitions of the processes $L_<(t)$ and $L_\leq(t)$
  8. Sources and sinks: stationarity
  9. Processes $L_<(t)$ and $L_\leq(t)$: non-asymptotic bounds
  10. Proof of Theorem \ref{['Th:Strict']} when $k_n\to +\infty$: de-Poissonization
  11. Proof of Theorem \ref{['Th:Large']}
  12. Proof for large $(k_n)$
  13. The gap between small and large $(k_n)$: Conclusion of the proof of Theorem \ref{['Th:Large']}
  14. Conclusion: Proof of Proposition \ref{['prop:cvProba']}
  15. Useful tail inequalities
  16. ...and 2 more sections

Key Result

Theorem 1

Let $(k_n)$ be a sequence of integers such that $k_n\leq n$ for all $n$. ThenIf $k_n\geq n$ for some $n$ then the following greedy strategy shows that $\mathbb{E}[\mathcal{L}_{<}(S_{k_n;n})]= n-\mathrm{o}(n)$ so the picture is complete. Indeed, first choose the leftmost point $(x_1,1)$ in $S_{k_n;n}

Figures (6)

  • Figure 1: A uniform $5$-multiset permutation $S_{5;30}$ of size $n=30$ and one of its longest non-decreasing subsequences.
  • Figure 2: The event $F$. (Ties are surrounded in red. Points with blue background represent the subsequence with $\delta\sqrt{n}$ ties.)
  • Figure 3: Our four variants of the Hammersley process (time goes from bottom to top, trajectories of particules are indicated in blue). Top left: The process $L_<(t)$. Top right: The process $L_\leq(t)$. Bottom left: The process $L^{(\alpha,p)}_{< }(t)$. Bottom right: The process $L^{(\beta,\beta^\star)}_{\leq }(t)$.
  • Figure 4: A sample of $\Pi^{(\lambda)}_{x,t}$, sources, sinks, and the corresponding trajectories of particles (in blue). Here $\mathcal{L}_{=<}(\Pi^{(\lambda)}_{x,t}\cup {\sf So}^{(\alpha)}_x\cup {\sf Si}^{(p)}_t)=5$ (pink path) and $L^{(\alpha,p)}_{< }(t)=2$ (two remaining particles at the top of the box).
  • Figure 5: Illustration of the notation of Lemma \ref{['lem:Valentin']}. Top: the multiset permutation $S_{k;n}$. Bottom: the corresponding $\widetilde{S}$. The longest non-decreasing subsequence in $S_{k;n}$ (circled points) is mapped onto a non-decreasing subsequence in $\widetilde{S}$, except one point with height $>A\lfloor n/A\rfloor$.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Theorem 1: Longest increasing subsequences
  • Theorem 2: Longest non-decreasing subsequences
  • Proposition 3
  • proof : Proof of Theorems \ref{['Th:Strict']} and \ref{['Th:Large']} in the case of a small sequence $(k_n)$
  • Remark
  • Proposition 4
  • proof
  • Lemma 5
  • Lemma 6
  • proof : Proof of Lemmas \ref{['lem:Stationnaire<']} and \ref{['lem:Stationnaire_leq']}
  • ...and 24 more