The absence of a pattern and the number of occurrences of another

TL;DR

This paper investigates the expected frequency of a permutation pattern within permutations avoiding another pattern, identifying extremal patterns and providing exact enumeration formulas, some of which are unexpected.

Contribution

It introduces new results on the expected occurrences of permutation patterns in r-avoiding permutations, including identifying least and most frequent patterns and deriving exact formulas.

Findings

Identified patterns with minimal and maximal occurrences in r-avoiding permutations

Derived exact enumeration formulas for pattern occurrences

Discovered some surprising enumeration results

Abstract

Following a question of J. Cooper, we study the expected number of occurrences of a given permutation pattern $q$ in permutations that avoid another given pattern $r$. In some cases, we find the pattern that occurs least often, (resp. most often) in all $r$-avoiding permutations. We also prove a few exact enumeration formulae, some of which are surprising.