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Hello I have a doubt about the next partition identities

$P(n: even \; number \; of \; odd \; parts)=P(n: distinct\; parts, \; number \; of \; odd\; parts\; is\; even)$

Also for the case when both instances of "even" are changed to odd.

I need to show a bijection of but I can't figure out how. I have been trying with one example (n=6) to see any particular characteristic however I can not have the equality.

I trying to solve the problem by using a bijection between the partitions of each side because I am not allowed to use function now.

So, any hint would be appreciated!

Liddo
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  • If $n$ is even, then all its partitions have an even number of odd parts. Therefore your identity as written would reduce to $P(n) = P(n: distinct parts)$, which is not true. – Michael Lugo Sep 14 '18 at 19:27
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    When you say $P(n:\text{even number of odd parts})$, I believe what you mean to say is $P(n:\text{even number of parts, and all parts are odd})$. As written, it sounds like $P(n:\text{even number of odd parts, any number of even parts})$ – Mike Earnest Sep 14 '18 at 19:56
  • And I guess when I chance "even" with "odd" I will have the same answer. I would like to know why in this website https://oeis.org/wiki/Partition_identities is like an identity. That is confusing. – Liddo Sep 14 '18 at 19:59
  • Here is a bijection: https://math.stackexchange.com/q/54961/177399 I do not understand what you are asking in your last comment. – Mike Earnest Sep 14 '18 at 20:03

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