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

TL;DR

This paper identifies permutation patterns where the count of permutations with a fixed number of pattern occurrences scales asymptotically with the total permutations avoiding that pattern, and explores the algebraic nature of related generating functions.

Contribution

It characterizes patterns with specific asymptotic behaviors and proves nonrationality and nonalgebraicity of certain generating functions for permutations with fixed pattern occurrences.

Findings

Asymptotic relation between permutations with fixed pattern occurrences and pattern avoidance

Identification of patterns with $r$ occurrences where counts scale as $n^r$ times avoiding permutations

Proof of nonrationality and nonalgebraicity of generating functions for these permutations

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.