Refinements of Beck-type partition identities

TL;DR

This paper refines Franklin's partition identity for the case j=1, proving it for fixed perimeter and deriving a Beck-type identity, with both analytic and combinatorial proofs.

Contribution

It provides a new refinement of Franklin's identity for j=1 and fixed perimeter, along with a Beck-type identity for partitions with fixed perimeter.

Findings

Proved Franklin's identity for j=1, r=2 with fixed perimeter.

Derived a Beck-type identity relating partitions with fixed perimeter.

Provided both analytic and combinatorial proofs.

Abstract

Franklin's identity generalizes Euler's identity and states that the number of partitions of $n$ with $j$ different parts divisible by $r$ equals the number of partitions of $n$ with $j$ repeated parts. In this article, we give a refinement of Franklin's identity when $j=1$. We prove Franklin's identity when $j=1$, $r=2$ for partitions with fixed perimeter, i.e., fixed largest hook. We also derive a Beck-type identity for partitions with fixed perimeter: the excess in the number of parts in all partitions into odd parts with perimeter $M$ over the number of parts in all partitions into distinct parts with perimeter $M$ equals the number of partitions with perimeter $M$ whose set of even parts is a singleton. We provide analytic and combinatorial proofs of our results.