I'm under the impression that if $n$ is even then the number of partitions of $n$ into an even number of parts exceeds the number of partitions of $n$ into an odd number of parts. And the opposite if $n$ is odd.
I feel like this is clear by looking at small examples, but I can't give a rigorous proof. I think it follows from how you make the partitions out of $n$ ones.
Would anyone be able to help me gain some more intuition about this or provide me with a proof? Thanks!