Number of (and way to enumerate) head to head schedules?

asked 2013-07-12 05:11:40 -0500

I have a fantasy football league with 8 members. The regular season of the league goes for 14 weeks, so each team plays the other seven teams exactly twice (once in a randomized order, and then over again in the same order). So for instance, if I am team 1, my list of opponents for the 14 weeks might look like this:


It is trivial to assess how many different schedules I as one player could face (7!, or 5,040). But for each of those schedules, there are some number of compatible schedules for the other players, which add up to a whole season schedule, and I am looking to find this number (the number of total schedules available for all players, not just one), and a way to generate these schedules in Excel. Any help would be greatly appreciated!

Notes: So far I have been able to assess that for any one sequence of opponents for one player, only 265 of 5040 sequences for a second player will be compatible. From there it gets dicey, since those 265*5040 combinations are not all isomorphic to one another.

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Is this a Sage question or a math question? That is, is it about implementing an algorithm, or is it about finding an algorithm in the first place?

John Palmieri gravatar imageJohn Palmieri ( 2013-07-14 05:39:34 -0500 )edit

I was thinking of it in both ways, but mostly in a mathematical way. That is, I'm interested in finding the answer, and if an easy (or at least straightforward) way to implement that answer falls out along with it, then that's a bonus. But I was more interested in a combinatorial way to find the answer (and possibly to generalize the answer for cases with N teams, where N is an even number, playing a schedule of N-1 weeks).

fargazmo gravatar imagefargazmo ( 2013-07-17 01:17:07 -0500 )edit