Classical pattern distributions in $\mathcal{S}_{n}(132)$ and $\mathcal{S}_{n}(123)$

TL;DR

This paper derives explicit recurrence relations for generating functions that count classical pattern occurrences within 132-avoiding and 123-avoiding permutations, advancing understanding of pattern distributions in these permutation classes.

Contribution

It introduces explicit recurrence relations for counting pattern occurrences in two important classes of pattern-avoiding permutations.

Findings

Derived recurrence relations for 132-avoiding permutations

Derived recurrence relations for 123-avoiding permutations

Enhanced understanding of pattern distribution in permutation classes

Abstract

Classical pattern avoidance and occurrence are well studied in the symmetric group $\mathcal{S}_{n}$. In this paper, we provide explicit recurrence relations to the generating functions counting the number of classical pattern occurrence in the set of 132-avoiding permutations and the set of 123-avoiding permutations.