29 citations · 154 across the 72 of their papers we have counts for
10 papers · 1 filter
The partial-fractions method for counting solutions to integral linear systems
Matthias Beck
We present a new tool to compute the number $ϕ_\A (\b)$ of integer solutions to the linear system $$ \x \geq 0 \qquad \A \x = \b $$ where the coefficients of $\A$ and $\b$ are inte…
The Number of Nowhere-Zero Flows on Graphs and Signed Graphs
Matthias Beck, Thomas Zaslavsky
A nowhere-zero -flow on a graph is a mapping from the edges of to the set such that, in any fixed orientation of , at eac…
Inside-Out Polytopes
Matthias Beck, Thomas Zaslavsky
We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and g…
The polynomial part of a restricted partition function related to the Frobenius problem
Matthias Beck, Ira M. Gessel, Takao Komatsu
Given a set of positive integers A = {a_1,...,a_n}, we study the number p_A (t) of nonnegative integer solutions (m_1,...,m_n) to m_1 a_1 + ... m_n a_n = t. We derive an explicit f…
Counting Lattice Points by means of the Residue Theorem
Matthias Beck
We use the residue theorem to derive an expression for the number of lattice oints in a dilated n-dimensional tetrahedron with vertices at lattice points on each coordinate axis an…
A Closer Look at Lattice Points in Rational Simplices
Matthias Beck
We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n+1 rational vertices, we u…
Higher-dimensional Dedekind sums and their bounds arising from the discrete diagonal of the n-cube
Matthias Beck, Sinai Robins, Shelemyahu Zacks
Higher-dimensional Dedekind sums are defined as a generalization of a recent 1-dimensional probability model of Dilcher and Girstmair to a d-dimensional cube. The analysis of the f…
The Reciprocity Law for Dedekind Sums via the constant Ehrhart coefficient
Matthias Beck
We obtain a new motivated proof of the reciprocity law for Dedekind sums by computing the constant coefficient of the Ehrhart polynomial for a rectangular triangle in two ways. On…
The volume of the 10th Birkhoff polytope
Matthias Beck, Dennis Pixton
The n'th Birkhoff polytope is the set of all doubly stochastic matrices, that is, those matrices with nonnegative real coefficients in which every row and column…
Refined upper bounds for the linear Diophantine problem of Frobenius
Matthias Beck, Shelemyahu Zacks
We study the Frobenius problem: given relatively prime positive integers a_1,...,a_d, find the largest value of t (the Frobenius number g(a_1,...,a_d)) such that m_1 a_1 + ... m_d…