PARTITIONS WEIGHTED BY THE NUMBER OF TWO TYPES OF PARTS

We confirm the conjecture proposed by ourselves and J. Love-joy that for all n > 9 p^′e^{(n)> p′}o⁽ⁿ⁾ holds, where p^′e^{(n) (respectively,p′}o(n)) is the number of partitions of n having an even (respectively, odd) number of odd parts larger than twice of the number of even parts. Moreover, we examine the connec-tions between the number of partitions weighted by the number of two types of parts and partition functions from the literature on the theory of partitions.