cp's OEIS Frontend

A369799 Number of binary relations R on [n] such that q(R) is a quasi-order and s(R) is a strict partial order (where q(R) and s(R) are defined below).

%I A369799 #10 Feb 03 2024 10:15:04
%S A369799 1,2,13,237,11590,1431913,424559959,292150780260,456213083587511,
%T A369799 1589279411184268465,12188163803127032036308,
%U A369799 203538148644721100472292979,7336995548182992341725851094195,566597426371900580541745092349604750,93154354372753215966288131247384428212545,32423220989898980232206367503220063835343283713
%N A369799 Number of binary relations R on [n] such that q(R) is a quasi-order and s(R) is a strict partial order (where q(R) and s(R) are defined below).
%C A369799 For a relation R on [n], let E = domain(R intersect R^(-1)) and let F = [n]\E.  Then q(R) := R intersect E X E and let s(R) := R intersect F X F.
%H A369799 E. Norris, <a href="https://www.researchgate.net/publication/225547994_The_structure_of_an_idempotent_relation">The structure of an idempotent relation</a>, Semigroup Forum, Vol 18 (1979), 319-329.
%F A369799 a(n) = Sum_{k=0..n} A369776(n,k) * 3^(k*(n-k)).
%t A369799 nn = 18; posets =Select[Import["https://oeis.org/A001035/b001035.txt", "Table"],
%t A369799    Length@# == 2 &][[All, 2]]; p[x_] := Total[posets Table[x^i/i!, {i, 0, 18}]]; Map[Total, (Map[Select[#, # > 0 &] &,Table[n!, {n, 0, nn}] CoefficientList[
%t A369799       Series[ p[Exp[ y  x] - 1]*p[ x], {x, 0, nn}], {x, y}]])*
%t A369799   Table[Table[3^(k (n - k)), {k, 0, n}], {n, 0, nn}]]
%Y A369799 Cf. A369778, A369776, A001035, A000798.
%K A369799 nonn
%O A369799 0,2
%A A369799 _Geoffrey Critzer_, Feb 01 2024