# [seqfan] Re: chess tournaments:

Svante Linusson linusson at matematik.su.se
Wed Apr 28 15:58:43 CEST 1999

```>%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 seems to me that this sequence is the same as sequence A007747

The score sequences can be thought of as the partitions (a_1,...,a_n)
of 2\binom{n}{2} of length at most n that is majorised by 2n,2n-2,2n-4,...,2,0.
That is,

f(n,k):=2n+2n-2+...+(2n-2k+2)-(a_1+a_2+...+a_k) \ge 0 for all k.

Now, the sequence 0=f(n,0),f(n,1),f(n,2),...,f(n,n)=0
is of the required type in A007747.  This establish a bijection.

ID Number: A007747
Sequence:  2,5,16,59,247,1111,5302,26376,135670
Name:      Nonnegative integer points (n_1,n_2,...,n_N) in polytope
n_0=n_{N+1}=0, 2n_i-(n_{i+1}+n_{i-1}) <= 2, n_i >= 0, i=1,...,N.
References ``Laughlin's wave functions, Coulomb gases and expansions of the
discriminant'', by P.
Di Francesco, M. Gaudin, C. Itzykson and F. Lesage, Int. Jour.
of Mod. Phys. A, Vol. 9, No.
24 (1994) 4257-4351.
Keywords:  nonn
Offset:    1
Author(s): P. Di Francesco [philippe at amoco.saclay.cea.fr]

/Svante

=========================================================
Svante Linusson
Matematiska Institutionen	tel. 08-16 14 34
Stockholms Universitet		fax. 08-612 67 17
S-10691 Stockholm		linusson at matematik.su.se
SWEDEN

```