On Increasing and Invariant Parking Sequences

TL;DR

This paper introduces and analyzes increasing parking sequences, a generalization of parking functions allowing cars of different sizes, and explores invariance properties with combinatorial characterizations and enumeration.

Contribution

It defines increasing parking sequences, establishes bijections to lattice paths, and investigates invariance properties with new characterizations and counting formulas.

Findings

Counted increasing parking sequences using lattice path bijections

Characterized invariance properties in parking sequences

Provided enumerative results for various classes of parking sequences

Abstract

The notion of parking sequences is a new generalization of parking functions introduced by Ehrenborg and Happ. In the parking process defining the classical parking functions, instead of each car only taking one parking space, we allow the cars to have different sizes and each takes up a number of adjacent parking spaces after a trailer $T$ parked on the first $z-1$ spots. A preference sequence in which all the cars are able to park is called a parking sequence. In this paper, we study increasing parking sequences and count them via bijections to lattice paths with right boundaries. Then we study two notions of invariance in parking sequences and present various characterizations and enumerative results.