Top Coefficients of the Denumerant

Abstract

For a given sequence $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\alpha)(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+ \ldots+ \alpha_Nx_N+ \alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\alpha)(t)$ is a quasipolynomial function of $t$ of degree $N$. In combinatorial number theory this function is known as the $\textit{denumerant}$. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\alpha)(t)$ as step polynomials of $t$. Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\alpha)(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a $\texttt{MAPLE}$ implementation will be posted separately.

Keywords

• denumerants
• ehrhart quasi-polynomials
• partitions
• polynomial-time algorithm
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]