An Identity for the Partition Function Involving Parts of k Different Magnitudes

TL;DR

This paper establishes a new identity for the partition function involving parts of k different magnitudes, linking it to the self convolution of the unrestricted partition function, with a combinatorial proof provided.

Contribution

It introduces a novel identity connecting partition functions with parts of different magnitudes to self convolutions, expanding understanding of partition theory.

Findings

Partition function with parts of k different magnitudes equals self convolution of unrestricted partition function.

Provides a combinatorial proof of the new identity.

Extends previous work by Merca on partition functions.

Abstract

Using previous work by Merca, we show the partition function involving parts of k different magnitudes, shifted by the triangular numbers, equals the self convolution of the unrestricted partition function. We also provide a combinatorial proof of this result.