An algebraic combinatorial approach to Sylvester's denumerant

Abstract

For a positive integer sequence $\boldsymbol{a}=(a_1, \dots, a_{N+1})$, Sylvester's denumerant $E(\boldsymbol{a}; t)$ counts the number of nonnegative integer solutions to $\sum_{i=1}^{N+1} a_i x_i = t$ for a nonnegative integer $t$. It has been extensively studied and a well-known result asserts that $E(\boldsymbol{a}; t)$ is a quasi-polynomial in $t$ of degree $N$. A milestone is Baldoni et al.'s polynomial algorithm in 2015 for computing the top $k$ coefficients when $k$ is fixed. Their development uses heavily lattice point counting theory in computational geometry. In this paper, we explain their work in the context of algebraic combinatorics and simplify their computation. Our work is based on constant term method, Barvinok's unimodular cone decomposition, and recent results on fast computation of generalized Todd polynomials. We develop the algorithm \texttt{CT-Knapsack}, together with an implementation in \texttt{Maple}. Our algorithm avoids plenty of repeated computations and is hence faster.