Polytope numbers and their properties

TL;DR

This paper explores polytope numbers, their relation to triangulations, and develops methods to express these numbers as sums of simplex numbers, enhancing understanding of polytope combinatorics.

Contribution

It introduces a detailed analysis of pointed triangulations and provides methods to represent polytope numbers as sums of simplex numbers.

Findings

Polytope numbers can be expressed as sums of simplex numbers.

Pointed triangulations are key to representing polytope numbers.

New methods for decomposing polytope numbers into simplex components.

Abstract

Polytope numbers for a polytope are a sequence of nonnegative integers that are defined by the facial information of a polytope. Every polygon is triangulable and a higher dimensional analogue of this fact states that every polytope is triangulable, namely, every polytope can be decomposed into simplexes. Thus it may be possible to represent polytope numbers by sums of simplex numbers. We analyzes a special type of triangulation, called pointed triangulation, and develops several methods to represent polytope numbers by sums of simplex numbers.