%I A007747
%S A007747 1,2,5,16,59,247,1111,5302,26376,135670,716542,3868142,21265884,
%T A007747 118741369,671906876,3846342253,22243294360,129793088770,
%U A007747 763444949789,4522896682789,26968749517543,161750625450884
%N A007747 Number of nonnegative integer points (p_1,p_2,...,p_n) in polytope defined 
               by p_0 = p_{n+1} = 0, 2p_i - (p_{i+1} + p_{i-1}) <= 2, p_i >= 0, 
               i=1,...,n. Number of score sequences in a chess tournament with n+1 
               players (with 3 outcomes for each game).
%C A007747 A correspondence between the points in the polytope and the chess scores 
               was found by Svante Linusson (linusson(AT)matematik.su.se):
%C A007747 The score sequences are partitions (a_1,...,a_n) of 2C(n,2) of length 
               <= n that are majorised by 2n,2n-2,2n-4,...,2,0; i.e. f(n,k) := 2n+2n-2+...+(2n-2k+2)-(a_1+a_2+...+a_k) 
               >= 0 for all k. The sequence 0=f(n,0),f(n,1),f(n,2),...,f(n,n)=0 
               is in the polytope. This establishes the bijection.
%D A007747 P. Di Francesco, M. Gaudin, C. Itzykson and F. Lesage, Laughlin's wave 
               functions, Coulomb gases and expansions of the discriminant, Int. 
               Jour. of Mod. Phys. A, Vol. 9, No. 24 (1994) 4257-4351.
%D A007747 P. A. MacMahon, Chess tournaments and the like treated by the calculus 
               of symmetric functions, Coll. Papers I, MIT Press, 344-375.
%H A007747 Jon Schoenfield, <a href="b007747.txt">Table of n, a(n) for n = 0..39</
               a>
%H A007747 Jon Schoenfield, <a href="a007747.txt">Comments on this sequence</a>
%H A007747 <a href="Sindx_To.html#tournament">Index entries for sequences related 
               to tournaments</a>
%F A007747 Schoenfield (see Comments link) gives a recursive method for computing 
               this sequence.
%e A007747 With 3 players the possible scores sequences are {{0,2,4}, {0,3,3}, {1,
               1,4}, {1,2,3}, {2,2,2}}.
%e A007747 With 4 players they are {{0,2,4,6}, {0,2,5,5}, {0,3,3,6}, {0,3,4,5}, 
               {0,4,4,4}, {1,1,4,6}, {1,1,5,5}, {1,2,3,6}, {1,2,4,5}, {1,3,3,5}, 
               {1,3,4,4}, {2,2,2,6}, {2,2,3,5}, {2,2,4,4}, {2,3,3,4}, {3,3,3,3}}.
%Y A007747 Cf. A000571, A047730, A064626, A064422.
%Y A007747 Sequence in context: A019589 A087949 A028333 this_sequence A107283 A059237 
               A104547
%Y A007747 Adjacent sequences: A007744 A007745 A007746 this_sequence A007748 A007749 
               A007750
%K A007747 nonn,nice
%O A007747 0,2
%A A007747 P. Di Francesco (philippe(AT)amoco.saclay.cea.fr), N. J. A. Sloane (njas(AT)research.att.com).
%E A007747 More terms from David W. Wilson (davidwwilson(AT)comcast.net)