  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.
