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arXiv · 2311.11677

Asymptotics of self-overlapping permutations

Sergey Kirgizov·Khaydar Nurligareev
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Abstract

In this work, we study the concept of self-overlapping permutations, which is related to the larger study of consecutive patterns in permutations. We show that this concept admits a simple and clear geometrical meaning, and prove that a permutation can be represented as a sequence of non-self-overlapping ones. The above structural decomposition allows us to obtain equations for the corresponding generating functions, as well as the complete asymptotic expansions for the probability that a large random permutation is (non-)self-overlapping. In particular, we show that almost all permutations are non-self-overlapping, and that the corresponding asymptotic expansion has the self-reference property: the involved coefficients count non-self-overlapping permutations once again. We also establish complete asymptotic expansions of the distributions of very tight non-self-overlapping patterns, and discuss the similarities of the non-self-overlapping permutations to other permutation building blocks, such as indecomposable and simple permutations, as well as their associated asymptotics.

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Sergey Kirgizov, Khaydar Nurligareev. 2023-11-20. Asymptotics of self-overlapping permutations. https://doi.org/10.1016/j.disc.2025.114400

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