%I A000571 M1189 N0459
%S A000571 1,1,1,2,4,9,22,59,167,490,1486,4639,14805,48107,158808,531469,
%T A000571 1799659,6157068,21258104,73996100,259451116,915695102,3251073303,
%U A000571 11605141649,41631194766,150021775417,542875459724,1972050156181
%N A000571 Number of different scores that are possible in an n-team round-robin 
               tournament.
%C A000571 A tournament is a complete graph with one arrow on each edge; the score 
               of a node is its out-degree; a(n) is number of different score sequences 
               when there are n nodes.
%D A000571 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000571 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000571 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 123, Problem 21.
%D A000571 J. W. Moon, Topics on Tournaments. Holt, NY, 1968, p. 68 (but table contains 
               errors).
%D A000571 T. V. Narayana and D. H. Best, Computation of the number of score sequences 
               in round-robin tournaments, Canad. Math. Bull., 7 (1964), 133-136 
               (but table contains errors).
%H A000571 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ScoreSequence.html">Link to a section of The World of Mathematics.</
               a>
%H A000571 <a href="Sindx_To.html#tournament">Index entries for sequences related 
               to tournaments</a>
%H A000571 C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, <a href="http:/
               /arXiv.org/abs/math.CO/0506334">On the x-rays of permutations</a>
%F A000571 Let f_1(T, E)=1 if T=E>=0, =0 else; f_n(T, E)=0 if T-E<C(n-1, 2), =Sum_{k=0..E} 
               f_{n-1}(T-E, k) else; then a(n)=Sum_{E=[ n/2 ]..n-1} f_n(C(n, 2), 
               E), n >= 2.
%F A000571 Nonnegative integer points (p_1, p_2, ..., p_n) in polytope p_0=p_{n+1}=0, 
               2p_i -(p_{i+1}+p_{i-1}) <= 1, p_i >= 0, i=1, ..., n.
%e A000571 a(3)=2, since either one node dominates [ 2,1,0 ] or each node defeats 
               the next [ 1,1,1 ].
%Y A000571 Cf. A007747.
%Y A000571 Sequence in context: A024427 A092920 A035053 this_sequence A077003 A046917 
               A159329
%Y A000571 Adjacent sequences: A000568 A000569 A000570 this_sequence A000572 A000573 
               A000574
%K A000571 nonn,nice,easy
%O A000571 0,4
%A A000571 N. J. A. Sloane (njas(AT)research.att.com).
%E A000571 a(11) corrected by Kenneth Winston (Aug 05 1978). More terms from David 
               W. Wilson (davidwwilson(AT)comcast.net).