Rectified Simplex Polytope Numbers

TL;DR

This paper explores the sequences of polytope numbers for r-rectified simplices, demonstrating they can be expressed as alternating sums of simplex numbers, extending combinatorial understanding of these polytopes.

Contribution

It introduces a novel expression for r-rectified simplex polytope numbers as alternating sums of simplex numbers, generalizing previous combinatorial results.

Findings

Sequences for r-rectified simplices are expressed as alternating sums of simplex numbers.

The approach generalizes inclusion-exclusion principles to polytope number sequences.

Provides a combinatorial framework for understanding polytope numbers of rectified simplices.

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. We focus on $r$-rectified simplices and show that the sequences for these polytopes can be written as alternating sums of simplex numbers analogous to the inclusion-exclusion given by the geometric process of $r$-rectification.