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A000568 Number of outcomes of unlabeled n-team round-robin tournaments.
(Formerly M1262 N0484)
19
1, 1, 1, 2, 4, 12, 56, 456, 6880, 191536, 9733056, 903753248, 154108311168, 48542114686912, 28401423719122304, 31021002160355166848, 63530415842308265100288, 244912778438520759443245824 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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, Apr 08 2001

REFERENCES

Cropper, Sebrina Ruth, "Ranking Score Vectors of Tournaments" (2011). All Graduate Reports and Creative Projects. Paper 91. Utah State University, School of Graduate Studies, http://digitalcommons.usu.edu/gradreports/91. - From N. J. A. Sloane, Jun 10 2012

R. L. Davis, Structure of dominance relations, Bull. Math. Biophys., 16 (1954), 131-140.

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).

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.

R. L. Davis, Structure of dominance relations, Bull. Math. Biophys., 16 (1954), 131-140. [Annotated scanned copy]

Brendan McKay, Combinatorial Data.

John W. Moon, Topics on tournaments, Holt, Rinehard and Winston (1968), see page 115.

J. W. Moon and M. Goldberg, On the composition of two tournaments, Duke Mathematical Journal, vol.37, no.2 (1970), pp.323-332. (subscription required)

J. W. Moon and M. Goldberg, On the composition of two tournaments, Duke Mathematical Journal 37.2 (1970): 323-332. [Annotated scans of pages 331 and 332 only]

N. J. A. Sloane, Annotated scan of John Moon's tables of tournaments on up to 6 nodes

N. J. A. Sloane, Illustration of first 5 terms

N. J. A. Sloane, A second Maple program for A000568

Eric Weisstein's World of Mathematics, Tournament

Index entries for sequences related to tournaments

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 ].

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

CROSSREFS

Cf. A006125 for the labeled analog, A051337.

Sequence in context: A158569 A020106 A099928 * A177921 A128648 A128646

Adjacent sequences:  A000565 A000566 A000567 * A000569 A000570 A000571

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic

STATUS

approved

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Last modified August 25 21:01 EDT 2016. Contains 275841 sequences.