On the restricted partition function

TL;DR

This paper derives formulas for the restricted partition function, which counts solutions to linear equations with nonnegative integer variables, including its polynomial component.

Contribution

It introduces new formulas for the restricted partition function and its polynomial part, advancing understanding of partition enumeration with linear constraints.

Findings

Formulas for the restricted partition function $p_{ extbf{a}}(n)$

Explicit expression for the polynomial part of the partition function

Enhanced methods for counting solutions to linear Diophantine equations

Abstract

For a vector $\mathbf a=(a_1,\ldots,a_r)$ of positive integers we prove formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ and its polynomial part.