| In this thesis we study two related notions of pattern avoidance. One is in set partitions sigma of [n] = {1,2...,n} which are families of nonempty subsets B1,...,B k whose disjoint union is [n], written sigma = B1/.../Bk |-- S. The other is in restricted growth functions (RGFs) which are words w = a1 a2...an of positive integers such that a1 = 1 and ai ≤ 1+max{ a1,...,ai--1} for i > 1. The concept of pattern avoidance is built on a standardization map st on an object O, be it a set partition or RGF, where st(O) is obtained by replacing the ith smallest integer with i. A set partition sigma will contain a pattern pi if sigma has a subpartition which standardizes to pi, and when sigma does not contain pi we say sigma avoids pi. Pattern avoidance in RGFs is defined similarly. This work is the study of the generating functions for Wachs and White's statistics on RGFs over the avoidance classes of set partitions and RGFs.;The first half of the thesis concentrates on set partitions. We characterize most of these generating functions for avoiding single and multiple set partitions of length three, and we highlight the longer pattern 14/2/3, a partition of [4], as its avoidance class has a particularly nice characterization. The second half of this thesis will present our results about the generating functions for RGF patterns, starting with those of length three. We find many equidistribution properties which we prove using integer partitions and the hook decomposition of Young diagrams. For certain patterns of any length we provide a recursive formula for their generating functions including the pattern 12... k. We finish this presentation by discussing the patterns 1212 and 1221 which have connections to noncrossing and nonnesting partitions, respectfully. We find connections to two-colored Motzkin paths and define explicit bijections between these combinatorial objects. |
