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Closed-Form Formula for the Partition Function and Related Functions

Alfredo Nader

Abstract

We develop a new closed-form arithmetic and recursive formula for the partition function and a generalization of Andrews' smallest parts (spt) function. Using the inclusion-exclusion principle, we additionally develop a formula for the not-relatively prime partition function (which counts the number of partitions that are not relatively prime). Moreover, we prove a theorem involving the greatest common divisor of partitions, which allows us to link partitions to prime numbers and lets us derive a formula for the relatively prime function. Lastly, we develop numerous new identities for Jordan's totient function of second order, Euler's totient function, and Dedekind's psi function.

Closed-Form Formula for the Partition Function and Related Functions

Abstract

We develop a new closed-form arithmetic and recursive formula for the partition function and a generalization of Andrews' smallest parts (spt) function. Using the inclusion-exclusion principle, we additionally develop a formula for the not-relatively prime partition function (which counts the number of partitions that are not relatively prime). Moreover, we prove a theorem involving the greatest common divisor of partitions, which allows us to link partitions to prime numbers and lets us derive a formula for the relatively prime function. Lastly, we develop numerous new identities for Jordan's totient function of second order, Euler's totient function, and Dedekind's psi function.
Paper Structure (11 sections, 21 theorems, 125 equations, 7 tables)

This paper contains 11 sections, 21 theorems, 125 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Theorem \ref{['th:part']}
  3. Closed-form and recursive formula for the partition function
  4. Two parts
  5. Three parts
  6. Four parts
  7. Generalization
  8. Generalized smallest parts function
  9. Relatively and not-relatively prime partition function
  10. Links to totient functions
  11. Further work

Key Result

Lemma 1.1

$\gcd(\lambda_1^n)=n$ and $p(n,1)=1$ for all $n\geq1$.

Theorems & Definitions (50)

  • Lemma 1.1
  • proof
  • Remark
  • Theorem 1.2
  • Remark
  • proof : Proof of Theorem \ref{['th:part']}
  • Remark
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • ...and 40 more