king chickens

[Brendan McKay]

I don't have time right now, so if someone who knows how to use nauty
wants to jump into this, here is how to do it:
1. Fetch the unlabelled tournaments from
      http://cs.anu.edu.au/~bdm/data/digraphs.html
2. For each unlabelled tournament compute the automorphism group size A
   and the number of king chickens K.
3. The answer is the sum of K*(n!/A) over the unlabelled tournaments.

That will get the answer up to n=10 fairly easily.  The next case n=11
is plausible but more effort as there are nearly a billion tournaments.

Incidentally, the definition of "tournament" in the entry is incomplete.
It has to add that exactly one of the directed edges u->v, v->u is
present for each pair u,v of distinct vertices.

Brendan.


* N. J. A. Sloane <njas at research.att.com> [061115 16:13]:
> can anyone extend this?
> 
> %I A123553
> %S A123553 1,2,12,104
> %N A123553 A "king chicken" in a tournament graph (a directed labeled graph on n nodes) is a player A who for any other player B either beats B directly or beats someone who beats B.  Sequence gives total number of king chickens in all 2^(n(n-1)/2) tourna
> %C A123553 H. G. Landau showed in 1951 that a player is a king chicken if and only if he has the highest score. There may be several king chickens in a tournament.
> %D A123553 S. B. Maurer, The king chicken theorems, Math. Mag., 53 (1980), 67-80.
> %e A123553 For n = 3 there are 8 tournaments: six of the form A beats B and C, and B beats C, with one king chicken (A), and two of the form A beats B beats C beats A, with three king chickens each (A or B or C), for a total of 6*1 + 2*3 = 12.
> %K A123553 nonn,more,new
> %O A123553 1,2
> %A A123553 njas, Nov 14 2006
> 
> Neil