Power Partitions and Semi-m-Fibonacci Partitions

TL;DR

This paper extends a recent identity linking Semi-Fibonacci partitions to partitions into powers of two with odd multiplicities, generalizing it to Semi-m-Fibonacci partitions and partitions into powers of m with multiplicities not divisible by m.

Contribution

It generalizes Andrews' identity to a broader class of partitions involving arbitrary m, connecting Semi-m-Fibonacci partitions with specific partitions into powers of m.

Findings

Extended the identity to Semi-m-Fibonacci partitions and partitions into powers of m.

Established a new combinatorial equivalence for these partition sets.

Provided a framework for further generalizations in partition theory.

Abstract

George Andrews recently proved a new identity between the cardinalities of the set of Semi-Fibonacci partitions and the set of partitions into powers of two with all parts appearing an odd number of times. This paper extends the identity to the set of Semi-$m$-Fibonacci partitions of $n$ and the set of partitions of $n$ into powers of $m$ in which all parts appear with multiplicity not divisible by $m$.