The combinatorics of $k$-marked Durfee symbols

Abstract

Andrews recently introduced k-marked Durfee symbols which are connected to moments of Dyson's rank. By these connections, Andrews deduced their generating functions and some combinatorial properties and left their purely combinatorial proofs as open problems. The primary goal of this article is to provide combinatorial proofs in answer to Andrews' request. We obtain a relation between k-marked Durfee symbols and Durfee symbols by constructing bijections, and all identities on k-marked Durfee symbols given by Andrews could follow from this relation. In a similar manner, we also prove the identities due to Andrews on k-marked odd Durfee symbols combinatorially, which resemble ordinary k-marked Durfee symbols with a modified subscript and with odd numbers as entries.