Restricted growth function patterns and statistics

Abstract

A restricted growth function (RGF) of length n is a sequence w = w_1 w_2 ... w_n of positive integers such that w_1 = 1 and w_i is at most 1 + max{w_1,..., w_{i-1}} for i at least 2. RGFs are of interest because they are in natural bijection with set partitions of {1, 2, ..., n}. RGF w avoids RGF v if there is no subword of w which standardizes to v. We study the generating functions sum_{w in R_n(v)} q^{st(w)} where R_n(v) is the set of RGFs of length n which avoid v and st(w) is any of the four fundamental statistics on RGFs defined by Wachs and White. These generating functions exhibit interesting connections with integer partitions and two-colored Motzkin paths, as well as noncrossing and nonnesting set partitions.