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Cycle Structure of Random Standardized Permutations

Author Aurélien Guerder

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ACM Subject Classification
  • Mathematics of computing → Permutations and combinations
  • Mathematics of computing → Combinatorics on words
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • Random permutations
  • primitive words
  • combinatorics
  • asymptotic distribution
  • method of moments

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Abstract

In this article, we study a model of random permutations, which we call random standardized permutations, and which is based on a sequence of i.i.d. random variables. This model generalizes other ones, such as the riffle shuffle and the major-index-biased permutations. We first establish an exact result on the joint distribution of the number of cycles of given lengths, involving the notion of primitive words. From this result, we obtain various convergence results, most of which are proved using the method of moments. We prove that the number of small cycles may have either a Poisson limit distribution, or a limit distribution given by a countable sum of independent geometric distributions. Then, we establish a limit distribution for large cycles, which is the Poisson-Dirichlet process. Finally, we prove a central limit theorem for the total number of cycles.

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Aurélien Guerder. Cycle Structure of Random Standardized Permutations. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.15

Author Details

Aurélien Guerder
  • IECL, Université de Lorraine, France

Funding

The author would like to thank the ORION program for its contribution to the funding of AG’s research internship. This work has benefited from a French government grant managed by the Agence Nationale de la Recherche with the reference ANR-20-SFRI-0009. The author acknowledges partial support from French Research Agency (ANR), through the LOUCCOUM project ANR-24-CE40-7809.

Acknowledgements

The author would like to express his sincere gratitude to his PhD supervisor Valentin Féray for his invaluable guidance and support.

References

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