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A000595
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Number of binary relations on n unlabeled points.
(Formerly M1980 N0784)
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45
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1, 2, 10, 104, 3044, 291968, 96928992, 112282908928, 458297100061728, 6666621572153927936, 349390545493499839161856, 66603421985078180758538636288, 46557456482586989066031126651104256, 120168591267113007604119117625289606148096, 1152050155760474157553893461743236772303142428672
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OFFSET
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0,2
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COMMENTS
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Number of orbits under the action of permutation group S(n) on n X n {0,1} matrices. The action is defined by f.M(i,j)=M(f(i),f(j)).
Equivalently, the number of digraphs on n unlabeled nodes with loops allowed but no more than one arc with the same start and end node. - Andrew Howroyd, Oct 22 2017
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 76 (2.2.30)
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sept. 15, 1955, pp. 14-22.
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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FORMULA
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a(n) = sum {1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...] / (1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[s_1, s_2, ...] = 2^sum {i, j>=1} (gcd(i, j)*s_i*s_j). - Christian G. Bower, Jan 05 2004
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EXAMPLE
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Non-isomorphic representatives of the a(2) = 10 relations:
{}
{1->1}
{1->2}
{1->1, 1->2}
{1->1, 2->1}
{1->1, 2->2}
{1->2, 2->1}
{1->1, 1->2, 2->1}
{1->1, 1->2, 2->2}
{1->1, 1->2, 2->1, 2->2}
(End)
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MATHEMATICA
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Join[{1, 2}, Table[CycleIndex[Join[PairGroup[SymmetricGroup[n], Ordered], Permutations[Range[n^2-n+1, n^2]], 2], s] /. Table[s[i]->2, {i, 1, n^2-n}], {n, 2, 7}]] (* Geoffrey Critzer, Nov 02 2011 *)
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[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v];
a[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
dinorm[m_]:=If[m=={}, {}, If[Union@@m!=Range[Max@@Flatten[m]], dinorm[m/.Apply[Rule, Table[{(Union@@m)[[i]], i}, {i, Length[Union@@m]}], {1}]], First[Sort[dinorm[m, 1]]]]];
dinorm[m_, aft_]:=If[Length[Union@@m]<=aft, {m}, With[{mx=Table[Count[m, i, {2}], {i, Select[Union@@m, #1>=aft&]}]}, Union@@(dinorm[#1, aft+1]&)/@Union[Table[Map[Sort, m/.{par+aft-1->aft, aft->par+aft-1}, {0}], {par, First/@Position[mx, Max[mx]]}]]]];
Table[Length[Union[dinorm/@Subsets[Tuples[Range[n], 2]]]], {n, 0, 3}] (* Gus Wiseman, Jun 17 2019 *)
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PROG
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(GAP) NSeq := function ( n ) return Sum(List(ConjugacyClasses(SymmetricGroup(n)), c -> (2^Length(Orbits(Group(Representative(c)), CartesianProduct([1..n], [1..n]), OnTuples))) * Size(c)))/Factorial(n); end; # Dan Hoey, May 04 2001
(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, 2*gcd(v[i], v[j]))) + sum(i=1, #v, v[i])}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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Still more terms from Dan Hoey, May 04 2001
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STATUS
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approved
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