3 research outputs found
Generalized parking function polytopes
A classical parking function of length is a list of positive integers
whose nondecreasing rearrangement satisfies . The convex hull of all parking
functions of length is an -dimensional polytope in , which
we refer to as the classical parking function polytope. Its geometric
properties have been explored in (Amanbayeva and Wang 2022) in response to a
question posed in (Stanley 2020). We generalize this family of polytopes by
studying the geometric properties of the convex hull of -parking
functions for , which we refer to as
-parking function polytopes. We explore connections between these
-parking function polytopes, the Pitman-Stanley polytope, and the
partial permutahedra of (Heuer and Striker 2022). In particular, we establish a
closed-form expression for the volume of -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.Comment: 29 pages, 3 figures, comments welcome