A366866 Number of binary relations R on [n] such that the transitive closure of R contains the identity relation.

1, 1, 7, 253, 39463, 24196201, 56481554827, 502872837857293, 17309567681965278223, 2333553047265268677638161, 1243013506394568266481053180947, 2629978323181659930952963974617537173, 22170279317365870690118601982232935268994583

Offset: 0

Author

Geoffrey Critzer, Oct 25 2023

Keywords

• nonn

Comments

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.

Links

• D. A. Gregory, S. Kirkland, and N. J. Pullman, Power convergent Boolean matrices, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.
• E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
• S. Schwarz, On the semigroup of binary relations on a finite set , Czechoslovak Mathematical Journal, 1970.

Crossrefs

Cf. A366252, A003030, A000798.

Programs

Mathematica

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]

Formula

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).