2026/07/16 by Charlie Hill, Ambrose Luo, Vu Trinh +1 · 1 citation
#math.CO
For b=(b1,…,bn)∈ℤ>0n, a b-parking function is a sequence (β1,…,βn) of positive integers whose nondecreasing rearrangement β1'≤β2'≤⋯≤βn' satisfies βi'≤ b1+⋯+bi. The b-parking-function polytope \mathfrakXn(b) is the convex hull of all b-parking functions of length n in ℝn. We prove that every lattice slice of \mathfrakXn(b), obtained by fixing one coordinate at an integer value, is itself a b'-parking-function polytope of one dimension less, with an explicit parameter vector b'; this yields a recursion for the number of lattice points of \mathfrakXn(b). We further show that every dilate of a b-parking-function polytope is a translate of another such polytope, that the number of lattice points is a polynomial function of b, and we deduce an explicit formula for the Ehrhart polynomial of \mathfrakXn(b) for arbitrary b as a finite sum indexed by draconian sequences, resolving a problem of Hanada, Lentfer, and Vindas-Meléndez; an equivalent formula was recently obtained, independently, by Liu and Thawinrak in a closely related setting. In the special case b=(a,b,…,b), we obtain an explicit closed form and a generating function for the Ehrhart polynomial. As an application, we classify magic positivity in the two-parameter family \mathfrakXn(a,b)=\mathfrakXn(a,b,…,b): the polytope \mathfrakXn(a,b) is magic positive if and only if (n,a,b)≠(2,1,1). Thus, we answer a problem posed by Ferroni and Higashitani for \mathfrakXn(a,b). Our result extends recent work of Liu and Zhang on partial permutahedra and leads us to conjecture that magic positivity holds for every \mathfrakXn(b) with n≥3.