On $k$-measures and Durfee squares of partitions

TL;DR

This paper provides a combinatorial proof linking the $k$-measure of partitions to Durfee squares, generalizing a recent identity and exploring properties of partitions with fixed $k$-measure.

Contribution

It offers a bijective combinatorial proof of a recent identity and generalizes the result for all $k$-measures, advancing understanding of partition properties.

Findings

Established a combinatorial proof of the identity

Generalized the result for all $k$-measures

Enhanced understanding of partition structures

Abstract

Recently, Andrews, Bhattacharjee and Dastidar introduced the concept of $k$-measure of an integer partition, and proved a surprising identity that the number of partitions of $n$ which have $2$-measure $m$ is equal to the number of partitions of $n$ with a Durfee square of side $m$. The authors asked for a bijective proof of this result and also suggested a further exploration of the properties of the number of partitions of $n$ which have $k$-measure $m$ for $k \geq 3$. In this note, we complete these tasks. That is, we obtain a short combinatorial proof of the result of Andrews, Bhattacharjee and Dastidar, and using this proof, we easily generalize this result for $k$-measures.