The combinatorics of k-marked Durfee symbols

TL;DR

This paper provides combinatorial proofs connecting k-marked Durfee symbols to Durfee symbols, resolving open problems and establishing bijections that derive Andrews' identities.

Contribution

It offers the first combinatorial proofs of Andrews' identities on k-marked Durfee symbols using bijections, linking them to Durfee symbols.

Findings

Established a bijection between k-marked Durfee symbols and Durfee symbols.

Derived all of Andrews' identities on k-marked Durfee symbols combinatorially.

Proved identities for k-marked odd Durfee symbols with modified parameters.

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.