chess tournaments:
David W. Wilson
wilson at cabletron.com
Wed Apr 28 16:24:40 CEST 1999
N. J. A. Sloane wrote:
> %I A047657
> %S A047657 1,2,5,16,59,247
> %N A047657 Number of score sequences in chess tournament with n players (with 3 outcomes for each game).
> %D A047657 P. A. MacMahon, Chess tournamemts and the like treated by the calculus of symmetric functions, Coll. P
> apers I, MIT Press, 344-375.
> %O A047657 0,3
> %K A047657 nonn,more,nice
> %A A047657 njas
> %Y A047657 This is probably the same as either A028333 or A007747.
> %e A047657 With 3 players the possible scores are 420, 411, 330, 321, 222.
>
> This is probably the same as either A028333 or A007747, maybe someone could check!
> It is not obvious from the descriptions.
> Perhaps simplest thing would be if someone extended this sequence
>
> MacMahon also discusses the case of tournaments where there
> are n players and in each of the C(n,2) games one player gets
> 3,2,1 or 0 points and the other player gets 0,1,2 or 3.
> How many different score sequences are there?
> This begins 1, 2, 8, 37 - more terms, someone?
>
> For 3 players the possible scores are
> 630 621 540 531 522 441 432 333
>
> NJAs
We can parameterize these sequences on the total number of pointsp awarded per game. In such a game, there are p+1
possible game
outcomes, namely (0,p), (1,p-1), ..., and (p,0). Let A_p(n) count
the number of score sequences for n players for n = 1, 2, 3, ....
The A047657 = A_2, and your other requested sequence is A_3.
I have computed the first few values for A_p, for 1 <= p <= 5.
p A_p
1 1,1,2,4,9,22,59,167,490,1486
2 1,2,5,16,59,247,1111,5302,26376
3 1,2,8,37,198,1178,7548
4 1,3,13,76,521,3996,32923
5 1,3,18,131,1111,10461
Notes:
A047657 should be indexed starting at 1.
A_1 is probably the same as A000571. Whereas A_1 has a recurrence,
it might be possible to work one out for all A_p.
A_2 is probably the same as A007747.
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