A 20 Integers 12 ( 2012 ) 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.