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A186081 Number of binary relations R on {1,2,...,n} such that the transitive closure of R is the trivial relation. 2
1, 1, 4, 144, 25696, 18082560, 47025585664, 450955726792704, 16260917603754029056, 2253010420928564535951360, 1219004114245442237742488879104, 2601909995433633381004133738019815424, 22040854392120341022554569447470527813779456 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
For n >= 2, a(n) is the number of strongly connected binary relations on [n]. - Geoffrey Critzer, Dec 04 2023
LINKS
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)
EXAMPLE
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.
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 *)
CROSSREFS
Sequence in context: A186418 A122747 A069135 * A138176 A203424 A055209
KEYWORD
nonn,nice
AUTHOR
Geoffrey Critzer, Feb 12 2011
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
STATUS
approved

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Last modified July 20 13:32 EDT 2024. Contains 374445 sequences. (Running on oeis4.)