Permutations with a fixed number of occurrences of a monotone pattern

TL;DR

This paper establishes bounds on permutations with a fixed number of a specific pattern and investigates the algebraic nature of their generating functions, revealing non-rationality and non-algebraicity results.

Contribution

It introduces a new upper bound relating permutations with fixed pattern occurrences to pattern-avoiding permutations and analyzes their generating functions.

Findings

Bound on permutations with fixed pattern occurrences

Generating function for fixed pattern copies is not rational for odd k

Generating function is not algebraic for even k

Abstract

We bound the number of permutations with a fixed number $r$ of $321 \ominus p_0$ patterns by a constant times the number of permutations which avoid $321 \ominus p_0$. We use this new upper bound to show that the ordinary generating function for permutations with $r$ copies of $k(k-1)...1$ is not rational for odd $k \geq 3$ and not algebraic for even $k \geq 3$.