1
$\begingroup$

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 $m$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.