On interior polytope number sequences

TL;DR

This paper explores the relationship between interior polytope number sequences and simplex number sequences, revealing that coefficients in their linear combination relate to the polytope's triangulation h-vector, and reversing these coefficients yields the interior sequence.

Contribution

It establishes a connection between the coefficients of polytope number sequences and the h-vector of a specific triangulation, and shows that reversing these coefficients produces the interior sequence.

Findings

Coefficients correspond to h-vector components of a triangulation.

Reversing coefficients yields the interior polytope sequence.

Provides a recursive method for expressing polytope numbers.

Abstract

Polytope numbers for a given polytope are an integer sequence defined by the combinatorics of the polytope. Recent work by H. K. Kim and J. Y. Lee has focused on writing polytope number sequences as sums of simplex number sequences. In addition, these works have given a process for writing the polytope number sequence in a recursive fashion by using the interior sequence for the various k-faces of the polytope, each viewed as a k-dimensional polytope. This paper shows that the coefficients of the linear combination of simplex number are the h-vector components for a certain type of triangulation of the polytope. In addition, reversing the order of the coefficients in the linear combination is shown to equal the interior polytope sequence for this polytope.