# Determinants with Bernoulli polynomials and the restricted partition function

## TL;DR

This paper explores determinants involving Bernoulli polynomials and their relationship with the restricted partition function, providing new insights into counting solutions to linear Diophantine equations.

## Contribution

It introduces two natural determinants with Bernoulli polynomials and connects them to the restricted partition function, offering novel analytical tools.

## Key findings

- Established connections between Bernoulli polynomial determinants and partition functions
- Derived formulas linking determinants to counting solutions of linear equations
- Enhanced understanding of the structure of restricted partition functions

## 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 study two natural determinants of order $rD$ with Bernoulli polynomials and we present connections with 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 $x_1\geq 0, \ldots, x_r\geq 0$.