Parking Function Polytopes

We extend the notion of parking function polytopes and study their geometric and combinatorial structure, including normal fans, face posets, and $h$-polynomials, as well as their connections to other classes of polytopes. To capture their combinatorial features, we introduce generalizations of ordered set partitions, called binary partitions and skewed binary partitions. Using properties of preorder cones, we characterize the skewed binary partitions that are in bijection with the cones of the normal fan of a parking function polytope. This description of the normal fan yields an explicit formula for the $h$-polynomials of simple parking function polytopes in terms of generalized Eulerian polynomials. Finally, we relate parking function polytopes to several well-known polytopes, leading to additional results, including formulas for their volumes and Ehrhart polynomials.

💡 Research Summary

This paper, “Parking Function Polytopes” by Fu Liu and Warut Tha Winrak, presents a comprehensive generalization and analysis of the geometric and combinatorial structure of parking function polytopes. The authors extend the definition beyond the classical case (where u = (1, 2, …, n)) to allow u to be any non-decreasing vector of nonnegative real numbers, significantly broadening the scope of study.

The central methodological innovation is the introduction of new combinatorial objects: binary partitions and skewed binary partitions of the set