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arXiv · 2603.19194

Luck and magic for Pitman-Stanley polytopes and parking functions

Nicolas Avila·Luis Ferroni·Alejandro H. Morales
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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.

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Nicolas Avila, Luis Ferroni, Alejandro H. Morales. 2026-03-19. Luck and magic for Pitman-Stanley polytopes and parking functions. https://arxiv.org/abs/2603.19194

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