I've got an elementary, combinatoric question: If the number n is odd, why is the number of even parts = the number of times where each part appears an even number of time = 0 ?
I mean: Of course, the claim is true (one can show this with any concrete example), but how can you show this in general?
Thanks for any hint!
Example: if n = 5
Partitions of 5: 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1
The number of times where only even parts appears is 0.
The number of times where each part appears an even number of times is also 0.
But: How can I show the claim generally?
Best Answer
An odd number cannot be partitioned solely into even parts. That's because a sum of even numbers is even.
Also if a partition has the property that each part occurs an even number of times, then the total of all parts of the same size is even, so the total of all parts must be even, and so cannot be odd.