A polytope related to empirical distributions, plane trees, parking functions, and the associahedron

TL;DR

This paper introduces a new n-dimensional polytope linked to various combinatorial structures, providing formulas for its volume and lattice points, and offering two polyhedral decompositions related to posets and the associahedron.

Contribution

It defines a novel polytope connected to multiple combinatorial objects and derives explicit formulas for its volume and lattice points, along with two polyhedral decompositions.

Findings

Explicit volume formulas for Pi_n(x)

Count of integer points in Pi_n(x) when x_i are integers

Polyhedral decompositions related to posets and the associahedron

Abstract

We define an n-dimensional polytope Pi_n(x), depending on parameters x_i>0, whose combinatorial properties are closely connected with empirical distributions, plane trees, plane partitions, parking functions, and the associahedron. In particular, we give explicit formulas for the volume of Pi_n(x) and, when the x_i's are integers, the number of integer points in Pi_n(x). We give two polyhedral decompositions of Pi_n(x), one related to order cones of posets and the other to the associahedron.