Coefficients of Sylvester's Denumerant

For a given sequence α = [α 1 , α 2 , . . . , α N +1 ] of N + 1 positive integers, we consider the combinatorial function E(α)(t) that counts the non-negative integer solutions of the equation α 1 x 1 + α 2 x 2 + · · · + α N x N + α N +1 x N +1 = t, where the right-hand side t is a varying non-negative integer. It is well-known that E(α)(t) is a quasi-polynomial function in the variable t of degree N . In combinatorial number theory this function is known as Sylvester's denumerant. Our main result is a new algorithm that, for every fixed number k, computes in polyno-mial time the highest k + 1 coefficients of the quasi-polynomial E(α)(t) as step polynomials of t (a simpler and more explicit representation). Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for E(α)(t) and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Our algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral cone into unimodular cones. This paper also presents a simple algorithm to predict the first non-constant coefficient and concludes with a report of several compu-tational experiments using an implementation of our algorithm in LattE integrale. We compare it with various Maple programs for partial or full computation of the denumerant.