Multi-dimensional versions of a formula of Popoviciu
Tóm tắt
In this paper, an explicit formulation for multivariate truncated power functions of degree one is given firstly. Based on multivariate truncated power functions of degree one, a formulation is presented which counts the number of non-negative integer solutions of s × (s + 1) linear Diophantine equations and it can be considered as a multi-dimensional versions of the formula counting the number of non-negative integer solutions of ax + by = n which is given by Popoviciu in 1953.
Tài liệu tham khảo
Pommersheim J. Toric varieties, lattice points, and Dedekind sums. Math Ann, 295(1): 1–24 (1993)
Schmidt J R, Bincer A. The Kostant partition function for simple Lie algebras. J Mathematical Physics, 25(8): 2367–2373 (1984)
Nijehuis A, Wilf H. Representation of integers by linear forms in nonnegative integers. J Number Theory, 4(1): 98–106 (1972)
Dinwoodie I. Stochastic simulation on integer constraint sets. SIAM J Optimization, 9(1): 953–961 (1999)
Welsh D. Approximate counting. In: Bailey R A, eds. Surveys in Combinatorics. Lecture Notes. London: London Mathematical Society, 1997, 241
Popoviciu T. Asupra unei probleme de patitie a numerelor. Acad Republicii Populare Romane, Filiala Cluj, Studii si cercetari stiintifice, 4: 7–58 (1953)
Beck M, Robins S. Computing the continuous discretely integer point enumeration in polyhedron. Springer Undergraduate Texts in Mathematics. In press
Sertöz S. On the number of solutions of the Diophantine equation of Frobenius. Diskret Math, 10(2): 62–71 (1988)
Tripathi A. The number of solutions to ax + by = n. Fibonacci Quart, 38(4): 290–293 (2000)
Dahmen W, Micchelli C A. The number of solutions to linear Diophantine equations and multivariate splines. Trans Amer Math Soc, 308(2): 509–532 (1988)
Dahmen W. On multivariate B-splines. SIAM J Numer Anal, 17(2): 179–191 (1980)
Jia R Q. Symmetric magic squares and multivariate splines. Linear Algebra Appl, 250(1–3): 69–103 (1997)
Xu Z. Discrete Truncated Powers, Volume of convex polytopes and Ehrhart polynomials. http://arxiv.org/abs/math.CO/0505129
Wang R, Xu Z. Multivariate splines and lattice points in rational polytope. J Comp Appl Math, 159(1): 149–159 (2003)
Dahmen W, Micchelli C A. On the solution of certain systems of partial difference equations and linear dependence of translates of box splines. Trans Amer Math Soc, 292(1): 305–320 (1985)