Permutations avoiding a pattern from Sk and at least two patterns from S3

In this paper, we find explicit formulas or generating functions for the cardinalities of the sets Sn(T, τ) of all permutations in Sn that avoid a pattern τ ∈ Sk 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 Sn(T, τ) for k = 4.