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Title: Generalized Parking Function Polytopes
Abstract

A classical parking function of lengthnis a list of positive integers$$(a_1, a_2, \ldots , a_n)$$(a1,a2,,an)whose nondecreasing rearrangement$$b_1 \le b_2 \le \cdots \le b_n$$b1b2bnsatisfies$$b_i \le i$$bii. The convex hull of all parking functions of lengthnis ann-dimensional polytope in$${\mathbb {R}}^n$$Rn, 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$${\textbf{x}}$$x-parking functions for$${\textbf{x}}=(a,b,\dots ,b)$$x=(a,b,,b), which we refer to as$${\textbf{x}}$$x-parking function polytopes. We explore connections between these$${\textbf{x}}$$x-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$${\textbf{x}}$$x-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.

 
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NSF-PAR ID:
10472917
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Annals of Combinatorics
Volume:
28
Issue:
2
ISSN:
0218-0006
Format(s):
Medium: X Size: p. 575-613
Size(s):
["p. 575-613"]
Sponsoring Org:
National Science Foundation
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