Restricted growth function patterns and statistics

TL;DR

This paper investigates the combinatorial properties of restricted growth functions (RGFs), their avoidance patterns, and related generating functions, revealing connections to integer partitions, Motzkin paths, and set partitions.

Contribution

It introduces new generating functions for RGFs avoiding specific patterns and explores their links to various combinatorial structures.

Findings

Generated functions relate to integer partitions and Motzkin paths

Established connections between RGFs and noncrossing, nonnesting partitions

Analyzed statistics on RGFs and their combinatorial implications

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.