How many arrangements are possible for a round robin tournament over an even number of players $n$?
A round robin tournament is a competition where $n = 2k$ players play each other once in a heads-up match (like the group stage of a FIFA World Cup). To accommodate this, there are $n-1$ rounds with $\frac{n}{2}$ games in each round. For an arrangement of a tournament, let's say that the matches within an individual round are unordered, but the rounds in the tournament are ordered. For $n$ players, how many possible arrangements of the tournament can there be?
…
I don't know if a formal statement is needed, but hey …
Let $P = \{ p_1, \ldots, p_n \}$ be a set of an even $n$ players. Let $R$ denote a round consisting of a set of pairs $(p_i,p_j)$ (denoting a match), such that $0<i<j\leq n$, and such that each player in $P$ is mentioned precisely once in $R$. Let $T$ be a tournament consisting of a tuple of $n-1$ valid rounds $(R_1, \ldots, R_{n-1})$, such that all rounds in $T$ are pair-wise disjoint (no round shares a match).
How many valid constructions of $T$ are there for $n$ input players?
The answer for 2 players is trivially 1. The answer for 4 players is 6. I believe the answer for 6 players to be 320. But how can this be solved in the general case?
Best Answer
This is almost the definition of a "$1$-factorization of $K_{2k}$", except that a $1$-factorization has an unordered set of matchings instead of a sequence of rounds. Since there are $2k-1$ rounds, this means that there are $(2k-1)!$ times as many tournaments, according to the definition above, as there are $1$-factorizations.
Counting $1$-factorizations of $K_{2k}$ seems to be a nontrival problem; see the Encyclopedia of Mathematics entry. The number of $1$-factorizations of $K_{2k}$ is OEIS sequence A000438. Also, see this paper (also here) for a count in the $k=7$ case.