number of different score sequences of size . The sequence starts as: 1, 1, 1, 2, 4, 9, 22, 59, 167, 490, 1486, 4639, 14805, 48107, ... Winston and Kleitman proved that for sufficiently large n: where Takács later showed, using some reasonable but unproven assumptions, that where Together these provide evidence that: Here signifies an asymptotically tight bound. Yao showed that every nonempty set of nonnegative integers is the score set for some tournament. Majority relations In social choice theory, tournaments naturally arise as majority relations of preference profiles. Let be a finite set of alternatives, and consider a list of linear orders over . We interpret each order as the preference ranking of a voter . The (strict) majority relation of over is then defined so that if and only if a majority of the voters prefer to , that is . If the number of voters is odd, then the majority relation forms the dominance relation of a tournament on vertex set . By a lemma of McGarvey, every tournament on vertices can be obtained as the majority relation of at most voters. Results by Stearns and Erdős & Moser later established that voters are needed to induce every tournament on vertices. Laslier (1997) studies in what sense a set of vertices can be called the set of "winners" of a tournament. This revealed to be useful in Political Science to study, in formal models of political economy, what can be the outcome of a democratic process. See also Oriented graph Paley tournament Sumner's conjecture Tournament solution Notes References . . . . . . . . . . . . . . . . . Directed graphs