Permutations Avoiding a Pattern from \(S_k\) and at Least two Patterns from \(S_3\)

Abstract

In this paper, we find explicit formulas or generating functions for the cardinalities of the sets \(S_n(T,\tau)\) of all permutations in \(S_n\) that avoid a pattern \(\tau \in S_k\) and a set \(T, |T| \geq 2,\) of patterns from \(S_3\). The main body of the paper is divided into three sections corresponding to the cases \(|T| = 2, 3\) and \(|T| \geq 4\). As an example, in the fifth section, we obtain the complete classification of all cardinalities of the sets \(S_n(T,\tau)\) for \(k = 4\).