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arXiv · 2403.07387

Combinatorics of generalized parking-function polytopes

Margaret M. Bayer·Steffen Borgwardt·Teressa Chambers·Spencer Daugherty·Aleyah Dawkins·Danai Deligeorgaki·Hsin-Chieh Liao·Tyrrell McAllister·Angela Morrison·Garrett Nelson·Andrés R. Vindas-Meléndez
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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 $(\beta_1,\dots,\beta_n)$ 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.

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Margaret M. Bayer, Steffen Borgwardt, Teressa Chambers, Spencer Daugherty, Aleyah Dawkins, Danai Deligeorgaki, Hsin-Chieh Liao, Tyrrell McAllister, Angela Morrison, Garrett Nelson, Andrés R. Vindas-Meléndez. 2024-03-12. Combinatorics of generalized parking-function polytopes. https://doi.org/10.1007/s00454-025-00770-1

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