Imagine you are given a set of $n$ (possibly with $n$ even) sport teams, and you want to explicitely construct all the possible fixture lists for a league in which each team has to play against all the other ones once, and each team has to play exactly one match contemporary to the other ones at each turn (it means that the league has to be played in $n-1$ "weeks"). How many of these combinations are there? And there is an easy iterative method to construct them?
Maybe the question have obvious solutions, but I wasn't able to find any of them for the moment. The only thing I know is that there are $(n-1)!!$ possible combinations for a single "week", but I have no idea of how to combine possible weeks to form an admissible schedule of $n-1$ turns.
I read already on Bipartite matching to construct schedule a condition to check if a given schedule is admissible, but I don't know if from that method is also possible to infer the exact number of possibilities, and a way to list them all.