Combinatorics of generalized parking-function polytopes

For \(\mathbf{b}=(b_1,\dots,b_n)\in \mathbb{Z}_{>0}^n\), a \(\mathbf{b}\)-parking function is defined to be a sequence \((\beta_1,\dots,\beta_n)\) of positive integers whose nondecreasing rearrangement \(\beta'_1\leq \beta'_2\leq \cdots \leq \beta'_n\) satisfies \(\beta'_i\leq b_1+\cdots + b_i\). The \(\mathbf{b}\)-parking-function polytope \(\mathfrak{X}_n(\mathbf{b})\) is the convex hull of all \(\mathbf{b}\)-parking functions of length \(n\) in \(\mathbb{R}^n\). Geometric properties of \(\mathfrak{X}_n(\mathbf{b})\) were previously explored in the specific case where \(\mathbf{b}=(a,b,b,\dots,b)\) and were shown to generalize those of the classical parking-function polytope. In this work, we study \(\mathfrak{X}_n(\mathbf{b})\) in full generality. We present a minimal inequality and vertex description for \(\mathfrak{X}_n(\mathbf{b})\), prove it is a generalized permutahedron, and study its \(h\)-polynomial. Furthermore, we investigate \(\mathfrak{X}_n(\mathbf{b})\) through the perspectives of building sets and polymatroids, allowing us to identify its combinatorial types and obtain bounds on its combinatorial and circuit diameters.