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

Combinatorial proofs and generalizations of conjectures related to Euler's partition theorem