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A000568
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Number of outcomes of unlabeled n-team round-robin tournaments.
(Formerly M1262 N0484)
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13
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1, 1, 1, 2, 4, 12, 56, 456, 6880, 191536, 9733056, 903753248, 154108311168, 48542114686912, 28401423719122304, 31021002160355166848, 63530415842308265100288, 244912778438520759443245824
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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REFERENCES
| R. L. Davis, Structure of dominance relations, Bull. Math. Biophys., 16 (1954), 131-140.
M. Goldberg and J. W. Moon, On the composition of two tournaments. Duke Math. J. 37 1970 323-332.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 157 and 523.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 126 and 245.
J. W. Moon, Topics on Tournaments. Holt, NY, 1968, p. 87.
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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Keith Briggs, Table of n, a(n) for n = 0..76
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Brendan McKay, Combinatorial Data.
N. J. A. Sloane, Annotated scan of John Moon's tables of tournaments on up to 6 nodes
N. J. A. Sloane, A second Maple program for A000568
Eric Weisstein's World of Mathematics, Tournament
Index entries for sequences related to tournaments
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FORMULA
| Davis's formula: a(n) = Sum_{j} (1/(Product (k^(j_k) (j_k)!))) * 2^{t_j},
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.,
and t_j = (1/2)*[ Sum_{r=1..n, s=1..n} j_r j_s gcd(r,s) - Sum_{r} j_r ].
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MAPLE
| 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:
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:
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:
od: print(n, s); od: (Vladeta Jovovic)
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CROSSREFS
| Cf. A006125 for the labeled analogue, A051337.
Sequence in context: A158569 A020106 A099928 * A177921 A128648 A128646
Adjacent sequences: A000565 A000566 A000567 * A000569 A000570 A000571
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs)
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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