The answer can be found in the "Encyclopedia of Integer Sequences", under the heading of "round-robin" tournaments.
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000571 I have reasonable confidence in this since the sequence matches my enumerations up to n = 7. Neil Henry Neil W. Henry wrote: > Josh wrote: > > >> If this is the case then we can apply a little graph theory to the >> problem. Each boat is a vertex/node/point of the graph and each race >> is represented by an edge/line/connection between each of the points. >> Each edge is directed, i.e. it points from, say, vertex A to B and not >> from B to A. Suppose that an edge directed from A to B means that boat >> A beat boat B in a race. Since each boat races every other boat in the >> group, there must be an edge connecting each pair of vertices. This is >> called a tournament graph (not a surprising name, eh?). >> >> This graph is explained in Eric's world of mathematics >> >> http://mathworld.wolfram.com/Tournament.html >> >> and the sequence of distinct graphs is given by >> >> >http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000568 >> >> >> Since there are only 9 syndicates in the Louis Vuitton Cup, the >> sequence given in the previous web page should answer your question. >> There also some references given if you want to find an extended >> sequence >> >> I hope this helps! >> >> Regards, >> >> >> Josh > > > While your description of the graph is correct, the results given in > Harary and Palmer, and on the att webpage you cite, are not solutions to > this particular problem. In other words, there is a theorem that says > for n = 5 there are exactly 12 "tournaments" of a certain type, but this > is not the number of possible standings asked for. > > My enumeration for n = 5 yields 9 possible standings (below). If I have > omitted one (or three!) please let me know. > > Neil Henry, Virginia Commonwealth University > > Standings for n = 5 round-robin competition: > 4 3 2 1 0 > 4 3 1 1 1 > 4 2 2 2 0 > 4 2 2 2 1 > 3 3 3 1 0 > 3 3 2 2 0 > 3 3 2 1 1 > 3 2 2 2 1 > 2 2 2 2 2 > . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
