On the $k$-measure of partitions and distinct partitions

TL;DR

This paper develops generating function identities for the $k$-measure of partitions and distinct partitions, extending previous results and providing new combinatorial insights into these partition statistics.

Contribution

It introduces trivariate generating functions for the length and $k$-measure of partitions and extends the $2$-measure case for partitions beyond prior work.

Findings

Derived new generating function identities for $k$-measure and length

Extended the $2$-measure case for partitions

Provided combinatorial formulas for partitions and distinct partitions

Abstract

The $k$-measure of an integer partition was recently introduced by Andrews, Bhattacharjee and Dastidar. In this paper, we establish trivariate generating function identities counting both the length and the $k$-measure for partitions and distinct partitions, respectively. The $2$-measure case for partitions extends a result of Andrews, Bhattacharjee and Dastidar.