Séminaire Lotharingien de Combinatoire, 95B.28 (2026), 12 pp.
Nicolas Avila, Luis Ferroni and Alejandro H. Morales
Luck aand Magic for Pitman-Stanley Polytopes aand Parking Functions
Abstract.
Pitman-Stanley polytopes are ubiquitous in geometric combinatorics. They appear naturally in statistical contexts and are tightly connected with the theory of parking functions in combinatorics. The present article addresses the lattice point enumeration of these polytopes, through the lens of Ehrhart theory. We give a combinatorial interpretation of the h*-polynomial of all Pitman-Stanley polytopes, and prove that they are real-rooted when the defining parameters of the polytope are positive. The crucial tool employed for this purpose is proving that Ehrhart polynomials of these Pitman-Stanley polytopes are magic positive. We achieve this by interpreting the coefficients of the Ehrhart polynomial in the magic basis via the combinatorics of "lucky" cars in a parking function.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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