On the restricted partition function II

TL;DR

This paper extends previous work by providing new formulas for the restricted partition function, which counts solutions to linear equations with non-negative integer variables, enhancing understanding of partition functions.

Contribution

It introduces additional formulas for the restricted partition function, advancing the theoretical framework established in prior research.

Findings

Derived new formulas for the restricted partition function

Enhanced the theoretical understanding of partition solutions

Built upon previous results to generalize formulas

Abstract

Let $\mathbf a=(a_1,\ldots,a_r)$ be a vector of positive integers. In continuation of a previous paper we present other 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 $x_1\geq 0, \ldots, x_r\geq 0$.