Application of multivariate splines to discrete mathematics

Details Application of multivariate splines to discrete mathematics Application of multivariate splines to discrete mathematics

2005-05-08

arxiv.org/abs/math/0505129v2

Mathematics

Combinatorics

Box splines; Number of integer points in polytopes; Pitman-Stanley polytope

Scientific paper

Using methods developed in multivariate splines, we present an explicit formula for discrete truncated powers, which are defined as the number of non-negative integer solutions of linear Diophantine equations. We further use the formula to study some classical problems in discrete mathematics as follows. First, we extend the partition function of integers in number theory. Second, we exploit the relation between the relative volume of convex polytopes and multivariate truncated powers and give a simple proof for the volume formula for the Pitman-Stanley polytope. Third, an explicit formula for the Ehrhart quasi-polynomial is presented.

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