My question is : is there a known method to count how many couple can we make out of $2n$ three members family ?
To be more precise, let's say that we have $6n$ individuals grouped three by three in families and we want to count how many ways there are to make couples out of them, such that,
- No couple are made out of two members of the same familly.
- Sex is irrelevant (there's no sex, or we don't care about that : it's a math problem).
- Every individual belongs to one and only one couple.
I've found this problem very hard to say the least. Maybe some version of the Wick theorem or the combinatorial theory of Hermite polynomials can help solveing it.
An answer where the individuals are initially indistinguishable (unlabeled enumeration) is highly preferred but any answer will be appreciated. Counting those coupling is in my humble opinion much easier if one consider that the individuals are numbered from $1$ to $6n$
I have an answer (both labeled and unlabeled) to count those couplings by generating functions without assumption 1. above. But I can't find a way to incorporate that condition.
p.s : this is my first post here.