On the partitions into distinct parts and odd parts

Abstract

In this paper, we show that the difference between the number of parts in the odd partitions of n and the number of parts in the distinct partitions of n satisfies Euler’s recurrence relation for the partition function p(n) when n is odd. A decomposition of this difference in terms of the total number of parts in all the partitions of n is also derived. In this context, we conjecture that for k > 0, the series