A003086
Number of self-complementary digraphs with n nodes.
Original entry on oeis.org
1, 1, 4, 10, 136, 720, 44224, 703760, 179228736, 9168331776, 9383939974144, 1601371799340544, 6558936236286040064, 3837878966366932639744, 62879572771326489528942592, 128777257564337108286016980992
Offset: 1
- F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 140, 243.
- R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
Table[GraphPolynomial[n,x,Directed]/.x -> -1,{n,1,20}] (* Geoffrey Critzer, Oct 21 2012 *)
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_] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[2 v[[i]] - 1, {i, 1, Length[v]}];
a[n_] := (s = 0; Do[s += permcount[2 p]*2^edges[p]*If[OddQ[n], n *4^Length[p], 1], {p, IntegerPartitions[n/2 // Floor]}]; s/n!);
Array[a, 16] (* Jean-François Alcover, Aug 26 2019, after Andrew Howroyd *)
-
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) = {4*sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, 2*v[i]-1)}
a(n) = {my(s=0); forpart(p=n\2, s+=permcount(2*Vec(p))*2^edges(p)*if(n%2, n*4^#p, 1)); s/n!} \\ Andrew Howroyd, Sep 16 2018
A005639
Number of self-converse oriented graphs with n nodes.
Original entry on oeis.org
1, 2, 5, 18, 102, 848, 12452, 265759, 10454008, 598047612, 63620448978, 9974635937844, 2905660724913768, 1268590412128132389, 1023130650177394611897, 1258149993547327488275562, 2834863110716120144290954314, 9900859865505110360978721901778
Offset: 1
- R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
- R. W. Robinson, Asymptotic number of self-converse oriented graphs, pp. 255-266 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Andrew Howroyd, Table of n, a(n) for n = 1..50 (terms 1..27 from R. W. Robinson)
- R. W. Robinson, Asymptotic number of self-converse oriented graphs, pp. 255-266 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978. (Annotated scanned copy)
- Sridharan, M. R., Self-complementary and self-converse oriented graphs , Nederl. Akad. Wetensch. Proc. Ser. A 73=Indag. Math. 32 1970 441-447. [Annotated scanned copy] See page 446.
-
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[If[Mod[v[[i]] v[[j]], 2] == 0, GCD[v[[i]], v[[j]]], 0], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[If[Mod[v[[i]], 2] == 0, 2 Quotient[v[[i]] - 2, 4] + 1, 0], {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Array[a, 18] (* Jean-François Alcover, Aug 17 2019, after Andrew Howroyd *)
-
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, if(v[i]*v[j]%2==0, gcd(v[i],v[j])))) + sum(i=1, #v, if(v[i]%2==0, (v[i]-2)\4*2+1))}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Sep 16 2018
A059735
Number of complementary pairs of tournaments on n nodes.
Original entry on oeis.org
1, 1, 2, 3, 10, 34, 272, 3528, 97144, 4870920, 452016608, 77054901728, 24271105072736, 14200712295904928, 15510501136026729216, 31765207922047709885696, 122456389219489134370435456, 891699423142395494501906828160, 12302820585630191716774996205431296
Offset: 1
- Andrew Howroyd, Table of n, a(n) for n = 1..50
- Nevena Francetić, Sarada Herke, Ian M. Wanless, Parity of Sets of Mutually Orthogonal Latin Squares, arXiv:1703.04764 [math.CO], 2017. See Section 4.1.
- V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.
-
A000568 = Cases[Import["https://oeis.org/A000568/b000568.txt", "Table"], {, }][[All, 2]];
A002785 = Cases[Import["https://oeis.org/A002785/b002785.txt", "Table"], {, }][[All, 2]];
a[n_] := (A000568[[n + 1]] + A002785[[n]])/2;
Array[a, 19] (* Jean-François Alcover, Aug 27 2019 *)
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 06 2005
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