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 A123553 A "king chicken" in a tournament graph (a directed labeled graph on n nodes with a single arc between every pair of 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) tournaments. 4
 1, 2, 12, 128, 2680, 109824, 8791552 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS H. G. Landau showed in 1951 that there may be several king chickens in a tournament and that a player is a king chicken if he has the highest score. The converse is not true and there can be more king chickens than highest scorers. The smallest example has 4 players: A beats B and C, B beats C and D, C beats D, D beats A; D is a king chicken despite having fewer points than A and B. Maurer showed in 1980 that there is one king chicken if one player beats all others and otherwise there are at least three. LINKS S. B. Maurer, The king chicken theorems, Math. Mag., 53 (1980), 67-80. FORMULA a(n) >= A006125(n)*3 - A006125(n-1)*n*2 with equality for n<=4. EXAMPLE 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. CROSSREFS Cf. A006125, A013976, A123553, A125032, A125031 (highest scorers)w, A123903 (Emperors). Sequence in context: A259267 A014235 A098628 * A354493 A079199 A303926 Adjacent sequences: A123550 A123551 A123552 * A123554 A123555 A123556 KEYWORD nonn,more AUTHOR N. J. A. Sloane, Nov 14 2006 EXTENSIONS Corrected and edited by Martin Fuller, Nov 16 2006 STATUS approved

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Last modified March 31 18:36 EDT 2023. Contains 361672 sequences. (Running on oeis4.)