The powers of two as sums over partitions

TL;DR

This paper explores two mathematical methods to express powers of two as sums over integer partitions, involving binomial coefficients and generating functions, and suggests potential for discovering additional methods.

Contribution

It introduces and compares two approaches for representing powers of two as sums over partitions, highlighting possible new methods.

Findings

Two methods for expressing powers of 2 as sums over partitions are analyzed.

Experimental results indicate the potential for additional decomposition methods.

The paper connects partition sums with binomial coefficients and generating functions.

Abstract

In this paper, we investigate two methods to express the natural powers of $2$ as sums over integer partitions. First we consider a formula by N. J. Fine that allows us to express a binomial coefficient in terms of multinomial coefficients as a sum over partitions. The second method invokes the central binomial coefficients and the logarithmic differentiation of their generating function. Some experimental results suggest the existence of other methods of decomposing the power of $2$ as sums over partitions.