2005/05/08 by Zhiqiang Xu, Xu, Zhiqiang
Computer Science · Engineering · Mathematics · #05A15 #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA) #Polynomial and algebraic computation #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.math/0505129
openalex publication_date 2005/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.