Journal ArticleGeneralized Parking Function PolytopesHanada, Mitsuki; Lentfer, John; Vindas-Meléndez, Andrés R<title>Abstract</title> <p>A classical parking function of length<italic>n</italic>is a list of positive integers<inline-formula><alternatives><tex-math>$$(a_1, a_2, \ldots , a_n)$$</tex-math><math><mrow><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>n</mi></msub><mo>)</mo></mrow></math></alternatives></inline-formula>whose nondecreasing rearrangement<inline-formula><alternatives><tex-math>$$b_1 \le b_2 \le \cdots \le b_n$$</tex-math><math><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>≤</mo><msub><mi>b</mi><mn>2</mn></msub><mo>≤</mo><mo>⋯</mo><mo>≤</mo><msub><mi>b</mi><mi>n</mi></msub></mrow></math></alternatives></inline-formula>satisfies<inline-formula><alternatives><tex-math>$$b_i \le i$$</tex-math><math><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>≤</mo><mi>i</mi></mrow></math></alternatives></inline-formula>. The convex hull of all parking functions of length<italic>n</italic>is an<italic>n</italic>-dimensional polytope in<inline-formula><alternatives><tex-math>$${\mathbb {R}}^n$$</tex-math><math><msup><mrow><mi>R</mi></mrow><mi>n</mi></msup></math></alternatives></inline-formula>, which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of<inline-formula><alternatives><tex-math>$${\textbf{x}}$$</tex-math><math><mi>x</mi></math></alternatives></inline-formula>-parking functions for<inline-formula><alternatives><tex-math>$${\textbf{x}}=(a,b,\dots ,b)$$</tex-math><math><mrow><mi>x</mi><mo>=</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></alternatives></inline-formula>, which we refer to as<inline-formula><alternatives><tex-math>$${\textbf{x}}$$</tex-math><math><mi>x</mi></math></alternatives></inline-formula>-parking function polytopes. We explore connections between these<inline-formula><alternatives><tex-math>$${\textbf{x}}$$</tex-math><math><mi>x</mi></math></alternatives></inline-formula>-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of<inline-formula><alternatives><tex-math>$${\textbf{x}}$$</tex-math><math><mi>x</mi></math></alternatives></inline-formula>-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.</p>Springer2024-06-0110633345Annals of Combinatorics282575 to 6130218-0006https://doi.org/10.1007/s00026-023-00671-12102921National Science Foundation