On the restricted partition function via determinants with Bernoulli polynomials

TL;DR

This paper establishes a determinant-based method involving Bernoulli polynomials to compute the restricted partition function, linking algebraic properties of determinants to combinatorial enumeration of integer solutions.

Contribution

It introduces a novel determinant criterion using Bernoulli polynomials for calculating the restricted partition function, providing explicit formulas under certain non-vanishing conditions.

Findings

Determinant  depends only on r and D

Restricted partition function expressed via Bernoulli polynomials

Explicit formulas involve Bernoulli Barnes numbers

Abstract

Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. We prove that, if a determinant $\Delta_{r,D}$, which depends only on $r$ and $D$, with entries consisting in values of Bernoulli polynomials is nonzero, then 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 $x_1\geq 0, \ldots, x_r\geq 0$ can be computed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers.