The score sequence of a tournament on graph vertices is the sequence
of their outdegrees listed in nondecreasing order,
where
is a positive integer. Elements of a score sequence
of length
therefore lie between 0 and , inclusively. Score sequences are so named because they
correspond to the multiset of scores obtained 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
outdegrees sum to , where is a binomial coefficient.
For example, the unique possible score sequence for is . For , the two possible score sequences are and . And for , the four possible score sequences are , , , and (OEIS A068029).
Landau (1953) has shown that a sequence of integers () is a score sequence
iff
The numbers 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.
For example, up to graph isomorphism there are
two 5-tournaments with score sequence and three with .
graph vertices is the sequence
of their outdegrees listed in nondecreasing order,
where 
is a positive integer. Elements of a score sequence
of length 
therefore lie between 0 and 
, inclusively. Score sequences are so named because they
correspond to the multiset of scores obtained 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
outdegrees sum to 
, where 
is a binomial coefficient.
is 
. For 
, the two possible score sequences are 
and 
. And for 
, the four possible score sequences are 
, 
, 
, and 
(OEIS A068029).
(
) is a score sequence
iff
,
..., 
,
where 
is a binomial coefficient, and equality for
, 2, ... are 1, 1, 2, 4, 9, 22, 59, 167, ... (OEIS A000571).
A score sequence does not uniquely determine a tournament.
For example, up to graph isomorphism there are
two 5-tournaments with score sequence 
and three with 
.
