0,4

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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 123, Problem 21.

J. W. Moon, Topics on Tournaments. Holt, NY, 1968, p. 68 (but table contains errors).

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).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to tournaments

C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the x-rays of permutations

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.

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.

a(3)=2, since either one node dominates [ 2,1,0 ] or each node defeats the next [ 1,1,1 ].

Cf. A007747.

Adjacent sequences: A000568 A000569 A000570 this_sequence A000572 A000573 A000574

Sequence in context: A024427 A092920 A035053 this_sequence A077003 A046917 A159329

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

a(11) corrected by Kenneth Winston (Aug 05 1978). More terms from David W. Wilson (davidwwilson(AT)comcast.net).