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Record-biased permutations and their permuton limit

Part of: Discrete mathematics in relation to computer science Combinatorics Combinatorial probability

Published online by Cambridge University Press:  27 March 2026

Mathilde Bouvel Open the ORCID record for Mathilde Bouvel [Opens in a new window] ,
Cyril Nicaud  and
Carine Pivoteau
Mathilde Bouvel*
Affiliation:
Université de Lorraine, CNRS, Inria, LORIA, Nancy, France
Cyril Nicaud
Affiliation:
Université Gustave Eiffel, CNRS, LIGM, Marne-la-Vallée, France
Carine Pivoteau
Affiliation:
Université Gustave Eiffel, CNRS, LIGM, Marne-la-Vallée, France
*
Corresponding author: Mathilde Bouvel; Email: mathilde.bouvel@loria.fr
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Abstract

In this article, we study a non-uniform distribution on permutations biased by their number of records that we call record-biased permutations. We give several generative processes for record-biased permutations, explaining also how they can be used to devise efficient (linear) random samplers. For several classical permutation statistics, we obtain their expectation using the above generative processes, as well as their limit distributions in the regime that has a logarithmic number of records (as in the uniform case). Finally, increasing the bias to obtain a regime with an expected linear number of records, we establish the convergence of record-biased permutations to a deterministic permuton, which we fully characterise. This model was introduced in our earlier work [3], in the context of realistic analysis of algorithms. We conduct here a more thorough study but with a theoretical perspective.

Keywords

Random permutationsnon-uniform distributionrecordsgenerative processrandom samplerpermutation statisticspermuton

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05A05: Permutations, words, matrices 68R01: General

Information

Type
Paper
Information
Combinatorics, Probability and Computing , First View , pp. 1 - 39
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Copyright
© The Author(s), 2026. Published by Cambridge University Press

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