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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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20
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1, 1, 1, 2, 4, 12, 56, 456, 6880, 191536, 9733056, 903753248, 154108311168, 48542114686912, 28401423719122304, 31021002160355166848, 63530415842308265100288, 244912778438520759443245824, 1783398846284777975419600287232, 24605641171260376770598003978281472
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OFFSET
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0,4
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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.
Cropper, Sebrina Ruth, Ranking Score Vectors of Tournaments (2011). All Graduate Reports and Creative Projects. Paper 91. Utah State University, School of Graduate Studies.
R. L. Davis, Structure of dominance relations, Bull. Math. Biophys., 16 (1954), 131-140. [Annotated scanned copy]
D. S. Dummit, E. P. Dummit, H. Kisilevsky, Characterizations of quadratic, cubic, and quartic residue matrices, arXiv preprint arXiv:1512.06480 [math.NT], 2015.
Robert A. Laird, Brandon S. Schamp, Calculating Competitive Intransitivity: Computational Challenges, The American Naturalist (2018), Vol. 191, No. 4, 547-552.
Robert A. Laird, Brandon S. Schamp, Exploring the performance of intransitivity indices in predicting coexistence in multispecies systems, Journal of Ecology (2018) Vol. 106, Issue 3, 815-825.
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]
Vladimír Müller, Jaroslav Nešetřil, Jan Pelant, Either tournaments or algebras?, Discrete Math. 11 (1975), 37-66. [Annotated copy] See table 1 on page 65.
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
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 11 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Tournament
Index entries for sequences related to tournaments
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FORMULA
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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
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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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MATHEMATICA
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permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i-1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]], 2], {i, 1, Length[v]}];
oddp[v_] := (For[i = 1, i <= Length[v], i++, If[BitAnd[v[[i]], 1] == 0, Return[0]]]; 1);
a[n_] := a[n] = (s = 0; Do[If[oddp[p] == 1, s += permcount[p]*2^edges[p]], {p, IntegerPartitions[n]}]; s/n!);
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 13 2017, after Andrew Howroyd *)
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PROG
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(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
oddp(v) = {for(i=1, #v, if(bitand(v[i], 1)==0, return(0))); 1}
a(n) = {my(s=0); forpart(p=n, if(oddp(p), s+=permcount(p)*2^edges(p))); s/n!} \\ Andrew Howroyd, Oct 22 2017
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CROSSREFS
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Cf. A006125 for the labeled analog, A051337.
Sequence in context: A158569 A020106 A099928 * A177921 A301481 A128648
Adjacent sequences: A000565 A000566 A000567 * A000569 A000570 A000571
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Vladeta Jovovic
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STATUS
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approved
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