Combinatorics of generalized parking-function polytopes

Abstract

For $\mathbf{b}=(b_1,\dots,b_n)\in \mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking function is defined to be a sequence $(β_1,\dots,β_n)$ of positive integers whose nondecreasing rearrangement $β'_1\leq β'_2\leq \cdots \leq β'_n$ satisfies $β'_i\leq b_1+\cdots + b_i$. 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.

Figures (5)

• Figure 1: The polytopes $\mathfrak{X}_3(1,2,3)$ (left) and $\mathfrak{X}_3(2,3,4)$ (right).
• Figure 2: The vertex poset $Q_{\overline{\mathbf{v}}}$ in \ref{['sublem: Vertex posets b1\n ge 2']}
• Figure 3: The vertex poset $Q_{\overline{\mathbf{v}}}$ in \ref{['sublem: Vertex posets b1 = 1']}
• Figure 4: The nestohedron from \ref{['ex:nestohedron']}.
• Figure 5: The combinatorial type of $\mathfrak{X}_{n}(\mathbf{b})$ when $n=3$ and $b_1\ge 2$.

Theorems & Definitions (59)

• Definition 1.1
• Definition 2.1
• Proposition 2.2
• proof
• Theorem 2.3
• proof : Proof of \ref{['thm:inequality_description']}
• Corollary 2.4
• proof
• Lemma 2.5
• proof