A366218 -id:A366218 - cp's OEIS frontend

A366252 Number of convergent binary relations on [n] (A365534) that converge to a quasi-order relation (A000798).

1, 1, 6, 227, 37617, 23750562, 56091061929

Offset: 0

Geoffrey Critzer, Oct 05 2023

Equivalently, a(n) is the number of convergent Boolean relation matrices whose Frobenius normal form is such that all the diagonal blocks are primitive (A070322).

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.
D. Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.

Cf. A070322, A365534, A000798, A366218.

nn = 6; B[n_] := 2^Binomial[n, 2] n!; pr[x_] := Total[primitive Table[x^i/i!, {i, 0, 6}]];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[- (pr[x] - 1)]], {x, 0, nn}], x]

Sum_{n>=0} a_n*x^n/(2^n*binomial(n,2)) = 1/(E(x) @ exp(-(p(x)-1))) where E(x) = Sum_{n>=0} x^n/(2^n*binomial(n,2)), p(x) is the e.g.f. for A070322, and @ is the exponential Hadamard product (see Panafieu and Dovgal).

A369397 Number of binary relations R on [n] such that the (unique) idempotent in {R,R^2,R^3,...} is an equivalence relation.

Original entry on oeis.org

1, 1, 5, 157, 26345, 18218521, 47136254765, 451286947588597, 16264532016440908625, 2253156851039460378774961, 1219026648017155982267265596885, 2601923405098893502520360223043594957, 22040885615442635622424409144799379027505465

Offset: 0

Geoffrey Critzer, Jan 22 2024

nonn

Equivalently, a(n) is the number of binary relations R on [n] such that the Frobenius normal form has no 0-blocks on the diagonal and all off diagonal blocks are 0-blocks.

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.
S. Schwarz, On the semigroup of binary relations on a finite set , Czechoslovak Mathematical Journal, 1970.

Cf. A366866 (binary relations R on [n] such that the (unique) idempotent in {R,R^2,R^3,...} is a quasiorder), A365534, A366218, A365590, A355612, A365593, A366252, A366350, A366218.

nn = 12; 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}]];
Table[n!, {n, 0, nn}] CoefficientList[Series[Exp [s[2 x] - x], {x, 0, nn}], x]

E.g.f.: exp(s(2x)-x) where s(x) is the e.g.f. for A003030.

cp's OEIS Frontend

A366252 Number of convergent binary relations on [n] (A365534) that converge to a quasi-order relation (A000798).

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