  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.

Combinatorics;Number Theory