Pattern Avoidance in Parking Functions

TL;DR

This paper studies pattern avoidance in parking functions by viewing them as labeled Dyck paths, enumerating those avoiding certain patterns, and establishing bijections with other combinatorial objects.

Contribution

It introduces a new perspective on parking functions as Dyck paths, enumerates pattern-avoiding cases, and finds bijections with various combinatorial structures.

Findings

Enumeration of parking functions avoiding sets of patterns of length 3

Identification of well-known combinatorial sequences in the counts

Bijections between pattern-avoiding parking functions and other combinatorial objects

Abstract

In this paper, we view parking functions viewed as labeled Dyck paths in order to study a notion of pattern avoidance first introduced by Remmel and Qiu. In particular we enumerate the parking functions avoiding any set of two or more patterns of length 3, and we obtain a number of well-known combinatorial sequences as a result. Along the way, we find bijections between specific sets of pattern-avoiding parking functions and a number of combinatorial objects such as partitions of polygons and trees with certain restrictions.