Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions

2026/08/03 by Feihu Liu, Jinlong Tang, Guoce Xin +1
Mathematics · #math.CO #math.NT

paper · pdf

19 pages

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

For a finite sequence of positive integers \boldsymbola=(a₁,…,a_n), the restricted partition function q_\boldsymbola(k) denote the number of nonnegative integer solutions to the equation a₁x₁+a₂x₂+⋯ +a_nx_n=k. It is proved to be a quasi-polynomial of degree n-1. Write q_\boldsymbola(k)=∑_j=0^n-1c_j(k)k^j with periodic coefficient functions c_j, and set b_m=#\i:m| a_i\. In 2008, Beck, Sam, and Woods conjectured that the minimum period of c_j(k) is lcm\m:b_m>j\. In this paper, we derive an exact root-of-unity formula for every coefficient function c_j(k). The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of c_j(k). Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.

Citations

• A polynomial time algorithm for Sylvester waves when entries are bounded
• An algebraic combinatorial approach to Sylvester's denumerant
• An Algebraic Approach to q-Partial Fractions and Sylvester Denumerants
• The Minimum Period of the Ehrhart Quasi-polynomial of a Rational\n Polytope