Combinatorial proofs and generalizations on conjectures related with Euler's partition theorem

Jane Y. X. Yang

Published 2018-01-21Version 1

Being aware of the recent work of Andrews, we notice two conjectures concerning with some variations of odd partitions and distinct partitions posed by Beck, which are analytically proved by Andrews. Later, following the same method of Andrews, Chern presented the analytic proof of another Beck's conjecture related 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.