The score sequence of a tournament is a monotonic nondecreasing sequence of the outdegrees of the graph vertices of the corresponding tournament graph. Elements of a score sequence of length therefore lie between 0 and , inclusively. Score sequences are so named because they correspond to the set of possible scores obtainable by the members of a group of players in a tournament where each player plays all other players and each game results in a win for one player and a loss for the other. (The score sequence for a given tournament is obtained from the set of outdegrees sorted in nondecreasing order, and so must sum to , where is a binomial coefficient.)
For example, the unique possible score sequences for is . For , the two possible sequences are and . And for , the four possible sequences are , , , and (OEIS A068029).
Landau (1953) has shown that a sequence of integers () is a score sequence iff
for , ..., , where is a binomial coefficient, and equality for
(Harary 1994, p. 211, Ruskey).
The number of distinct score sequences for , 2, ... are 1, 1, 2, 4, 9, 22, 59, 167, ... (OEIS A000571). A score sequence does not uniquely determine a tournament since, for example, there are two 4-tournaments with score sequence and three with .