Pattern avoiding partitions, sequence A054391, and the kernel method

Sequence A054391 in OEIS, which we will denote by a n , counts a certain two-pattern avoidance n class of the permutations of size n. In this paper, we provide additional combinatorial interpretations for these numbers in terms of finite set partitions. In particular, we identify six classes of the partitions of size n, all of which have cardinality a n and each avoiding two n classical patterns. We use both algebraic and combinatorial methods to establish our results. In one apparently more difficult case, to show the result, we make use of the kernel method in solving a system of three functional equations which arises after a certain parameter is introduced. We also define an algorithmic bijection between the avoidance class in this case and another which systematically replaces the occurrences of a given pattern with those of another having the same length.