Combinatorics of Generalized Parking-Function Polytopes

For b=(b₁, ,bₙ)∈ Z>0ⁿ b = ( b 1 , ⋯ , b n ) ∈ Z > 0 n , a b b -parking function is defined to be a sequence (β ₁, ,β ₙ) ( β 1 , ⋯ , β n ) of positive integers whose nondecreasing rearrangement β '₁≤ β '₂≤ ⋯ ≤ β 'ₙ β 1 ′ ≤ β 2 ′ ≤ ⋯ ≤ β n ′ satisfies β 'ᵢ≤ b₁+⋯ + bᵢ β i ′ ≤ b 1 + ⋯ + b i . The b b -parking-function polytope Xₙ(b) X n ( b ) is the convex hull of all b b -parking functions of length n in Rⁿ R n . Geometric properties of Xₙ(b) X n ( b ) were previously explored in the specific case where b=(a,b,b, ,b) b = ( a , b , b , ⋯ , b ) and were shown to generalize those of the classical parking-function polytope. In this work, we study Xₙ(b) X n ( b ) in full generality. We present a minimal inequality and vertex description for Xₙ(b) X n ( b ) , prove it is a generalized permutahedron, and study its h -polynomial. Furthermore, we investigate Xₙ(b) X n ( b ) through the perspectives of building sets and polymatroids, allowing us to identify its combinatorial types and obtain bounds on its combinatorial and circuit diameters.