A186081 Number of binary relations R on {1,2,...,n} such that the transitive closure of R is the trivial relation.

1, 1, 4, 144, 25696, 18082560, 47025585664, 450955726792704, 16260917603754029056, 2253010420928564535951360, 1219004114245442237742488879104, 2601909995433633381004133738019815424, 22040854392120341022554569447470527813779456

Offset: 0

Author

Geoffrey Critzer, Feb 12 2011

Keywords

• nonn
• nice

Comments

For n >= 2, a(n) is the number of strongly connected binary relations on [n]. - Geoffrey Critzer, Dec 04 2023

Examples

			a(2)=4 because there are four relations on {1,2} whose transitive closure is {(1,1), (1,2), (2,1), (2,2)}. They are the three nontransitive relations,{(1,2), (2,1)}, {(1,2), (2,1), (2,2)}, {(1,1), (1,2), (2,1)} and the trivial relation itself.
		

Crossrefs

Cf. A002416, A003030, A070322, A361269.

Programs

Mathematica

Needs["Combinatorica`"];
f[list_] := Apply[Plus, Table[MatrixPower[list,n], {n,1,Length[list]}]];
Join[{1}, Table[Length[Select[Map[Flatten, Map[f, Tuples[Strings[{0, 1}, n], n]]], FreeQ[#, 0] &]], {n, 1, 4}]]
(* Second program: *)
a[ n_] := If[ n < 1, Boole[n == 0], With[{triv = matnk[n, 2^n^2 - 1]}, Sum[ Boole[triv === transitiveClosure[ matnk[n, k]]], {k, 0, 2^n^2 - 1}]]]; matnk[n_, k_] := Partition[ IntegerDigits[ k, 2, n^2], n]; transitiveClosure[x_] := FixedPoint[ Sign@(# + Dot[#, x]) &, x, Length@x]; (* Michael Somos, Mar 08 2012 *)

Formula

From Geoffrey Critzer, Dec 04 2023: (Start)

For n >= 2, a(n) = A003030(n)*2^n = A361269(n,1).

E.g.f.: 1 + s(2*x) - x where s(x) is the e.g.f. for A003030. (End)

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 31 2015

a(6) from Bert Dobbelaere, Feb 16 2019

a(7)-a(12) from Geoffrey Critzer, Dec 04 2023