Inversions in parking functions

TL;DR

This paper develops generating functions and bijections to analyze inversions in parking functions, extends results to unit interval parking functions, and introduces probabilistic formulas for inversion statistics across various word sets.

Contribution

It introduces a q-exponential generating function for inversions in parking functions and applies symmetric functions, bijections, and probabilistic methods to analyze these structures.

Findings

Derived a q-exponential generating function for inversions

Established a bijection to rooted labeled forests

Provided formulas for total inversions and related statistics

Abstract

In this paper, we obtain a q-exponential generating function for inversions on parking functions via symmetric function theory and also through a direct bijection to rooted labeled forests. We then apply these techniques to unit interval parking functions to give analogous results. We conclude by introducing a probabilistic approach through which we obtain formulas for the total number of inversions and several other statistics across all parking functions and other sets of words closed under rearrangement.