On k-measures and Durfee squares of partitions

Abstract

Recently, Andrews, Bhattacharjee and Dastidar introduced the concept of [Formula: see text]-measure of an integer partition, and proved a surprising identity that the number of partitions of [Formula: see text] which have [Formula: see text]-measure [Formula: see text] is equal to the number of partitions of [Formula: see text] with a Durfee square of side [Formula: see text]. 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 [Formula: see text] which have [Formula: see text]-measure [Formula: see text] for [Formula: see text]. In this paper, we perform these tasks. That is, we obtain a short combinatorial proof of the result of Andrews, Bhattacharjee and Dastidar, and using this proof, we obtain a natural generalization for [Formula: see text]-measures.