Permutations avoiding a pattern from S_kand at least two patterns from S₃

In this paper, we find explicit formulas or generating functions for the cardinalities of the sets S_n(T, τ) of all permutations in S _n that avoid a pattern τ ∈ S_k and a set T, |T| ≥ 2, of patterns from S₃. The main body of the paper is divided into three sections corresponding to the cases |T| = 2, 3 and |T| ≥ 4. As an example, in the fifth section, we obtain the complete classification of all cardinalities of the sets S_n(T, τ) for k = 4.