A121244 Number of score vectors for tournaments on n nodes that do not determine the tournament uniquely.
0, 0, 0, 0, 2, 11, 41, 136, 437, 1397, 4490, 14554, 47683, 158093, 530265, 1797631, 6153650, 21252343, 73986392, 259434758, 915667537, 3251026851, 11605063370, 41631062856, 150021553132, 542875085143, 1972049524959
Offset: 1
Keywords
Examples
For n = 3 there are two possible score sequences: {0,1,2} and {1,1,1}. Both of them uniquely define the corresponding tournament. Hence a(3) = 0.
The first occurrence of a sequence that doesn't define a tournament happens for n = 5. There are two such sequences {1,1,2,3,3} and {1,2,2,2,3}. Let's consider the first sequence: {1,1,2,3,3}. Let's take the two best players - the persons with 3 wins - as one of them should win the game with another, there is only one other person who won a game with one of the two best players. It could happen that this player has score 1 or 2. Thus we can get two different tournaments with the same score vector.
Links
- Eric Weisstein's World of Mathematics, Score Sequence.