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A366866
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Number of binary relations R on [n] such that the transitive closure of R contains the identity relation.
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3
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1, 1, 7, 253, 39463, 24196201, 56481554827, 502872837857293, 17309567681965278223, 2333553047265268677638161, 1243013506394568266481053180947, 2629978323181659930952963974617537173, 22170279317365870690118601982232935268994583
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OFFSET
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0,3
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COMMENTS
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Equivalently, a(n) is the number of n X n Boolean relation matrices whose Frobenius normal form contains no 0-blocks on the diagonal. See Gregory, Kirkland, and Pullman.
Equivalently, a(n) is the number of labeled directed graphs on [n] (with self loops allowed) such that every strongly connected component contains at least one arc.
This sequence is a good upper-bound for the number of relations that converge to a quasi-order (A366252) which is only known for n <= 6.
If the transitive closure of a relation R contains the identity relation then there is exactly one transitive relation in {R,R^2,R^3...}. See Schwarz link.
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LINKS
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FORMULA
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Sum_{n>=0} a_n*x^n/(2^n*binomial(n,2)) = 1/(E(x) @ exp(-(s(2x)-x))) where E(x) = Sum_{n>=0} x^n/(2^n*binomial(n,2)), s(x) is the e.g.f. for A003030, and @ is the exponential Hadamard product (see Panafieu and Dovgal).
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MATHEMATICA
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nn = 12; B[n_] := 2^Binomial[n, 2] n!; strong = Select[Import["https://oeis.org/A003030/b003030.txt", "Table"], Length@# == 2 &][[All, 2]]; s[x_] := Total[strong Table[x^i/i!, {i, 1, 58}]]; ggf[egf_]:=Normal[Series[egf, {x, 0, nn}]] /.Table[x^i ->x^i/2^Binomial[i, 2], {i, 0, nn}]; Table[B[n], {n, 0, nn}] CoefficientList[
Series[1/ggf[Exp[- (s[2 x] - x)]], {x, 0, nn}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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