A formula for the number of partitions of $n$ in terms of the partial Bell polynomials

TL;DR

This paper presents a novel formula expressing the partition number p(n) using partial Bell polynomials, combining Faà di Bruno's formula and Euler's pentagonal number theorem.

Contribution

It introduces a new explicit formula for p(n) in terms of partial Bell polynomials, linking combinatorial partition theory with Bell polynomial identities.

Findings

Derived a formula for p(n) using partial Bell polynomials.

Connected partition numbers with Faà di Bruno's formula.

Utilized Euler's pentagonal number theorem in the derivation.

Abstract

We derive a formula for $p(n)$ (the number of partitions of $n$) in terms of the partial Bell polynomials using Fa\`{a} di Bruno's formula and Euler's pentagonal number theorem.