Luck and magic for Pitman-Stanley polytopes and parking functions

TL;DR

This paper proves a positivity property for Ehrhart polynomials of Pitman--Stanley polytopes, linking combinatorics of parking functions to geometric and algebraic properties like real-rootedness.

Contribution

It introduces the concept of magic positivity for Ehrhart polynomials and provides a combinatorial interpretation involving lucky cars in parking protocols.

Findings

Ehrhart polynomials exhibit magic positivity.

h*-polynomials are real-rooted, log-concave, and unimodal.

Combinatorial interpretation via lucky cars in parking functions.

Abstract

Motivated by the combinatorics of parking functions and their several generalizations, we study the Ehrhart theory of Pitman--Stanley polytopes. We prove a strong positivity phenomenon called \emph{magic positivity} for the Ehrhart polynomials of these polytopes, which in turn implies that their $h^*$-polynomials are real-rooted (and thus log-concave and unimodal). Our result is achieved by interpreting the coefficients of these Ehrhart polynomials in the \emph{magic basis} in terms of the number of \emph{lucky cars} in a modified parking protocol. Furthermore, we address the magic positivity problem for $\mathbf{y}$-generalized permutohedra and also discuss a \emph{magic} combinatorial interpretation for them, under the assumption that the input parameters are sufficiently large.