Avoiding type (1,2) or (2,1) patterns in a partition of a set

A partition π of the set (n) = {1,2, . . . , n} is a collection {B1, . . . , Bk} of nonempty pairwise disjoint subsets of (n) (called blocks) whose union equals (n). In this paper, we find exact formulas and/or generating functions for the number of partitions of (n) with k blocks, where k is fixed, which avoid 3-letter patterns of type x − yz or xy − z, providing generalizations in several instances. In the particular cases of 23 − 1, 22 − 1, and 32 − 1, we are only able to find recurrences and functional equations satisfied by the generating function, since in these cases there does not appear to be a simple explicit formula for it.