Formulas for the number of $k$-colored partitions and the number of plane partitions of $n$ in terms of the Bell polynomials

TL;DR

This paper derives explicit formulas for counting k-colored partitions and plane partitions of n using Bell polynomials, providing new combinatorial enumeration tools.

Contribution

It introduces closed-form formulas expressing partition counts in terms of Bell polynomials, advancing combinatorial enumeration methods.

Findings

Formulas for k-colored partitions in terms of Bell polynomials

Formulas for plane partitions in terms of Bell polynomials

Enhanced tools for combinatorial enumeration

Abstract

We derive closed formulas for the number of $k$-coloured partitions and the number of plane partitions of $n$ in terms of the Bell polynomials.