Department of Mathematical Sciences, KAIST, Korea
Polytope numbers for a polytope are a sequence of nonnegative integers which are defined by the facial information of a polytope. This is a higher dimensional generalization of polygonal number. It is well known that every polygon can be decomposed into triangles. A higher dimensional analogue of this fact states that every polytope has a triangulation, namely, it can be decomposed into simplices. Thus it may be possible to represent polytope numbers as sums of simplex numbers, which gives another way of calculating polytope numbers.
In this talk, we define polytope numbers and calculate polytope numbers for several polytopes, and we introduce decomposition theorem, which is a way of representing polytope numbers as sums of simplex numbers. We also suggest further problems in the study of polytope numbers and possible approaches to these problems.
Joint work with Prof. Hyun Kwang Kim, POSTECH, Korea.