Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mircea Cimpoeas

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 $0\leq i\leq m-1$.

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

Abstract

We prove new formulas and congruences for the number of partitions of into parts and the number of partitions of into distinct parts. Also, we give lower and upper bounds for the density of the set , where and .

Paper Structure

This paper contains 6 sections, 11 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. The polynomial part of $p(n,k)$ and $q(n,k)$
  5. The Sylvester waves of $p(n,k)$ and $q(n,k)$
  6. New formulas and congruences for $p(n,k)$ and $q(n,k)$

Key Result

Proposition 3.1

We have that where $f_{k,m}(n) = d_{\mathbf k,m}(n-k),\text{ and }\mathbf k=(1,2,\ldots,k)$.

Theorems & Definitions (21)

  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • Proposition 4.1
  • Theorem 4.2
  • ...and 11 more