A note on the log-concavity of parking functions

TL;DR

This paper proves the log-concavity of a specific statistic on parking functions, extending the results to G-parking functions and connected graphs, using recent matroid log-concavity results.

Contribution

It settles Bóna's conjecture on parking functions' log-concavity and generalizes the results to G-parking functions and connected graphs.

Findings

Confirmed the log-concavity of a parking function statistic.

Extended log-concavity results to G-parking functions.

Proved log-concavity for connected, labeled graphs graded by edges.

Abstract

We settle a conjecture of B\'ona regarding the log-concavity of a certain statistic on parking functions by utilizing recent log-concavity results on matroids. This result allows us to also prove that connected, labeled graphs graded by their number of edges are log-concave. Furthermore, we generalize these results to $G$-parking functions.