Permutations with a fixed number of occurrences of a pattern: A case generalizing 231

Michael Waite

Permutations with a fixed number of occurrences of a pattern: A case generalizing 231

Combinatorics 2026-03-24 v1

Authors: Michael Waite

Abstract

We determine a set of permutation patterns $q$ so that the number of permutations with $r$ occurrences of $q$ is asymptotically $n^r$ times the number of permutations avoiding $q$ , partially settling a conjecture of Conway and Guttman. We also use these asymptotics to prove nonrationality and nonalgebraicity for certain ordinary generating functions for permutations with $r$ copies of a pattern.

Keywords

permutation patterns pattern occurrence asymptotic enumeration generating functions nonrationality nonalgebraicity Conway-Guttman conjecture

Cite

@article{arxiv.2603.21625,
  title  = {Permutations with a fixed number of occurrences of a pattern: A case generalizing 231},
  author = {Michael Waite},
  journal= {arXiv preprint arXiv:2603.21625},
  year   = {2026}
}

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10 pages, 2 figures

Permutations with a fixed number of occurrences of a pattern: A case generalizing 231

Abstract

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