# Partitions of $n$ into even number of parts versus into odd number of parts

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!

Your conjecture is correct for $$n>2$$.
If $$n>2$$, $$n$$ has at least one self-conjugate partition; see the answer to this question. If $$c(n)$$ is the number of self-conjugate partitions of $$n$$, $$c(n)=(-1)^n(p_E(n)-p_O(n))\,,$$ where $$p_E(n)$$ and $$p_O(n)$$ are the numbers of partitions of $$n$$ with an even and an odd number of parts, respectively; this is proved in the answers to this question. Thus, $$c(n)$$ is positive for $$n>2$$, so $$p_E(n)-p_O(n)$$ must be positive for even $$n>2$$ and negative for odd $$n>2$$.