compositions of n into even parts
By Emma Terry •
I have found here { that the number of integer compositions of n into k odd parts would be ${\frac{n+k-1}{2} \choose k-1}$.
I would like to find the number of integer compositions of n into k even parts. My guess is that it would be the same, but I do not see how to prove it.
$\endgroup$1 Answer
$\begingroup$Each positive, even number is at least equal to two, so you can subtract one to get an odd, but still positive, number.
Therefore, the number of ways to decompose $n$ into $k$ even numbers is the same as the number of compositions of $n-k$ into $k$ odd numbers; and for that you already have a formula.
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