# Permutations of a Fantasy Football Schedule

I am trying to determine the number of unique Fantasy Football schedules for the regular season. The constraint is that everyone must play each other roughly the same amount of times.

## Terminology

Suppose there are $n$ teams ($n$ must be even), and $m$ weeks.

The constraint is that no team can play another team more than $\frac{m}{n - 1}$ (rounded up) times. For example, if $n = 4$ and $m = 14$, then Team 1 cannot play Team 2 more than $5$ times.

### Schedules

Each schedule is composed of a list of weeks.

Order is important. Two schedules with the same set of weeks, but the weeks are in a different order, are unique schedules

### Weeks

Each week is composed of a set of matchups

The number of mathcups in a week is

$$\frac{m}{2} = \frac{number \quad of \quad teams}{2}$$ Order of matchups in a week is irrelevant.

### Matchups

Each matchup is a head to head matchup with two distinct teams

For the purpose of this question, a matchup between Team 1 and Team 2 is denoted as $(1,2)$

## What I Done Already

I have figured out how this works for a 4-team league ($n = 4$). This is essentially a permutation of a multiset.

For example, if $n = 4$ and $m = 14$

$$\# of Permutations = \frac{14!}{5!5!4!} * 3 = 756 756$$

## Where the Problem Gets Interesting

For the 4-team league ($n = 4$), there are only $3$ possible weeks

$$[(1,2),(3,4)] \\ [(1,3),(2,4)] \\ [(1,4),(2,3)] \\$$

Every week has unique matchups. (i.e. The matchup Team 1 vs. Team 2 only occurs in one week, etc.)

However, lets look at a 6-team league ($n = 6$). There are $15$ possible weeks

$$[(1,2),(3,4),(5,6)] \\ [(1,2),(3,5),(4,6)] \\ [(1,2),(3,6),(4,5)] \\ . \\ [(1,3),(2,4),(5,6)] \\ [(1,3),(2,5),(4,6)] \\ [(1,3),(2,6),(4,5)] \\ . \\ . \\ . \\$$

Now it's not just a simple multiset calculation, because different weeks will have the same matchup. And there is a constraint that every team must play each other roughly the same amount of time.

Example:

For $n = 6$ and $m = 6$, lets consider the following schedule

$$[(1,2),(3,4),(5,6)] \\ [(1,2),(3,5),(4,6)] \\ [(1,2),(3,6),(4,5)] \\ [(1,3),(2,4),(5,6)] \\ [(1,3),(2,5),(4,6)] \\ [(1,3),(2,6),(4,5)] \\$$

Every week is unique, but this is an invalid schedule because Team 1 plays Team 2 three times. However the maximum repeat matchups for $n = 6$ and $m = 6$ is $2$.

## The Question

1. I am mostly interested in the cases where $4 \leqslant n \leqslant 14$ ($n$ is even) and $10 \leqslant m \leqslant 17$.

2. It would be interesting to know if there is a general solution that works for all even $n$ and all $m$