  In a recent work, Andrews gave analytic proofs of two conjectures concerning
some variations of two combinatorial identities between partitions of a
positive integer into odd parts and partitions into distinct parts discovered
by Beck. Subsequently, using the same method as Andrews, Chern presented the
analytic proof of another Beck's conjecture relating the gap-free partitions
and distinct partitions with odd length. However, the combinatorial
interpretations of these conjectures are still unclear and required. In this
paper, motivated by Glaisher's bijection, we give the combinatorial proofs of
these three conjectures directly or by proving more generalized results.
