A note on the number of partitions of $n$ into $k$ parts

TL;DR

This paper introduces new formulas and congruences for counting partitions of integers into a fixed number of parts, along with bounds on the density of certain residue classes, advancing understanding of partition functions.

Contribution

It provides novel formulas and congruences for partition counts and establishes bounds on the density of residue classes for these functions.

Findings

New formulas for p(n,k) and q(n,k)

Congruences for partition functions

Bounds on the density of residue classes

Abstract

We prove new formulas and congruences for $p(n,k):=$ the number of partitions of $n$ into $k$ parts and $q(n,k):=$ the number of partitions of $n$ into $k$ distinct parts. Also, we give lower and upper bounds for the density of the set $\{n\in\mathbb N\;:\;p(n,k)\equiv i(\bmod\; m)\}$, where $m\geq 2$ and $0\leq i\leq m-1$.