|
%I #6 Mar 08 2023 10:14:52
%S 0,0,0,0,2,11,41,136,437,1397,4490,14554,47683,158093,530265,1797631,
%T 6153650,21252343,73986392,259434758,915667537,3251026851,11605063370,
%U 41631062856,150021553132,542875085143,1972049524959
%N Number of score vectors for tournaments on n nodes that do not determine the tournament uniquely.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ScoreSequence.html">Score Sequence</a>.
%F This sequence is the difference between A000571 (Number of different scores that are possible in an n-team round-robin tournament) and A000570 (Number of tournaments on n nodes determined by their score vectors).
%e 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.
%e 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.
%K nonn
%O 1,5
%A _Tanya Khovanova_, Aug 22 2006
|
