An algebraic combinatorial approach to Sylvester's denumerant

2023/12/04 by Guoce Xin, Chen Zhang, Xin, Guoce +1 · 1 citation
Computer Science · Mathematics · #05A17 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Primary 05--08 #Secondary 05--04

paper · pdf · doi:10.48550/arxiv.2312.01569

openalex publication_date 2023/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a positive integer sequence \boldsymbola=(a₁, …, a_N+1), Sylvester's denumerant E(\boldsymbola; t) counts the number of nonnegative integer solutions to ∑_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(\boldsymbola; 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 CT-Knapsack, together with an implementation in Maple. Our algorithm avoids plenty of repeated computations and is hence faster.

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