Search: id:A000568
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%I A000568 M1262 N0484
%S A000568 1,1,1,2,4,12,56,456,6880,191536,9733056,903753248,154108311168,
%T A000568 48542114686912,28401423719122304,31021002160355166848,
%U A000568 63530415842308265100288,244912778438520759443245824
%N A000568 Number of outcomes of unlabeled n-team round-robin tournaments.
%D A000568 R. L. Davis, Structure of dominance relations, Bull. Math. Biophys., 
               16 (1954), 131-140.
%D A000568 M. Goldberg and J. W. Moon, On the composition of two tournaments. Duke 
               Math. J. 37 1970 323-332.
%D A000568 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 157 and 523.
%D A000568 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, pp. 126 and 245.
%D A000568 J. W. Moon, Topics on Tournaments. Holt, NY, 1968, p. 87.
%D A000568 K. B. Reid and L. W. Beineke "Tournaments", pp. 169-204 in L. W. Beineke 
               and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic 
               Press, NY, 1978.
%D A000568 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000568 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000568 Keith Briggs, <a href="b000568.txt">Table of n, a(n) for n = 0..76</a>
%H A000568 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000568 Brendan McKay, <a href="http://cs.anu.edu.au/~bdm/data/digraphs.html">
               Combinatorial Data</a>.
%H A000568 N. J. A. Sloane, <a href="a000568.pdf">Annotated scan of John Moon's 
               tables of tournaments on up to 6 nodes</a>
%H A000568 N. J. A. Sloane, <a href="a000568.txt">A second Maple program for A000568</
               a>
%H A000568 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Tournament.html">Tournament</a>
%H A000568 <a href="Sindx_To.html#tournament">Index entries for sequences related 
               to tournaments</a>
%F A000568 Davis's formula: a(n) = Sum_{j} (1/(Product (k^(j_k) (j_k)!))) * 2^{t_j},
%F A000568 where j runs through all partitions of n into odd parts, say with j_1 
               parts of size 1, j_3 parts of size 3, etc.,
%F A000568 and t_j = (1/2)*[ Sum_{r=1..n, s=1..n} j_r j_s gcd(r,s) - Sum_{r} j_r 
               ].
%p A000568 with(combinat):with(numtheory): for n from 1 to 30 do p:=partition(n): 
               s:=0:for k from 1 to nops(p) do ex:=1:for i from 1 to nops(p[k]) 
               do if p[k][i] mod 2=0 then ex:=0:break:fi:od:
%p A000568 if ex=1 then q:=convert(p[k],multiset): for i from 1 to n do a(i):=0:od:for 
               i from 1 to nops(q) do a(q[i][1]):=q[i][2]:od:
%p A000568 c:=1:ord:=1:for i from 1 to n do c:=c*a(i)!*i^a(i): if a(i)<>0 then ord:=lcm(ord,
               i):fi:od: g:=0:for d from 1 to ord do if ord mod d=0 then g1:=0:for 
               del from 1 to n do if d mod del=0 then g1:=g1+del*a(del):fi:od:g:=g+phi(ord/
               d)*g1*(g1-1):fi:od: s:=s+2^(g/ord/2)/c:fi:
%p A000568 od: print(n,s); od: (Vladeta Jovovic)
%Y A000568 Cf. A006125 for the labeled analogue, A051337.
%Y A000568 Sequence in context: A158569 A020106 A099928 this_sequence A128648 A128646 
               A155747
%Y A000568 Adjacent sequences: A000565 A000566 A000567 this_sequence A000569 A000570 
               A000571
%K A000568 nonn,nice,easy
%O A000568 0,4
%A A000568 N. J. A. Sloane (njas(AT)research.att.com).
%E A000568 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs)
%E A000568 Harary and Palmer give incorrect values for a(24) and a(25); the correct 
               values are a(24) = 195692027657521876084316842660833482785173437775365039898624 
               and a(25) = 131326696677895002131450257709457767457170027052967027982788816896. 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 08 2001

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