Permutations containing a pattern exactly once and avoiding at least two patterns of three letters

TL;DR

This paper derives explicit formulas and recurrences for counting permutations that contain a specific pattern exactly once while avoiding multiple patterns of length three, expanding understanding of permutation pattern avoidance.

Contribution

It provides new explicit formulas and recurrence relations for permutations with a single occurrence of a pattern and multiple pattern avoidances, categorized by the size of the avoided set.

Findings

Formulas for |T|=2,3, and ≥4 cases

Recurrence relations for permutation counts

Enhanced understanding of pattern containment and avoidance

Abstract

In this paper, we find an explicit formulas, or recurrences, in terms of generating functions for the cardinalities of the sets $S_n(T;\tau)$ of all permutations in $S_n$ that contain $\tau\in S_k$ exactly once and avoid a subset $T\subseteq S_3$, $|T|\geq2$. The main body of the paper is divided into three sections corresponding to the cases $|T|=2,3$ and $|T|\geq4$.