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%I #43 Dec 06 2023 09:02:00
%S 1,1,4,144,25696,18082560,47025585664,450955726792704,
%T 16260917603754029056,2253010420928564535951360,
%U 1219004114245442237742488879104,2601909995433633381004133738019815424,22040854392120341022554569447470527813779456
%N Number of binary relations R on {1,2,...,n} such that the transitive closure of R is the trivial relation.
%C For n >= 2, a(n) is the number of strongly connected binary relations on [n]. - _Geoffrey Critzer_, Dec 04 2023
%F From _Geoffrey Critzer_, Dec 04 2023: (Start)
%F For n >= 2, a(n) = A003030(n)*2^n = A361269(n,1).
%F E.g.f.: 1 + s(2*x) - x where s(x) is the e.g.f. for A003030. (End)
%e 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.
%t f[list_] := Apply[Plus, Table[MatrixPower[list,n], {n,1,Length[list]}]]; Table[Length[Select[Map[Flatten, Map[f, Tuples[Strings[{0, 1}, n], n]]], FreeQ[#, 0] &]], {n, 0, 4}]
%t 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 *)
%Y Cf. A002416, A003030, A070322, A361269.
%K nonn,nice
%O 0,3
%A _Geoffrey Critzer_, Feb 12 2011
%E a(0)=1 prepended by _Alois P. Heinz_, Aug 31 2015
%E a(6) from _Bert Dobbelaere_, Feb 16 2019
%E a(7)-a(12) from _Geoffrey Critzer_, Dec 04 2023
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