Avoidance of partitions of a three-element set

Klazar defined and studied a notion of pattern avoidance for set partitions, which is an analogue of pattern avoidance for permutations. Sagan considered partitions which avoid a single partition of three elements. We enumerate partitions which avoid any family of partitions of a 3-element set as was done by Simion and Schmidt for permutations. We also consider even and odd set partitions. We provide enumerative results for set partitions restricted by generalized partition patterns, which are an analogue of the generalized permutation patterns of Babson and Steingrímsson. Finally, in the spirit of work done by Babson and Steingrímsson, we will show how these generalized partition patterns can be used to describe set partition statistics.