A361455
Triangle read by rows: T(n,k) is the number of simple digraphs on labeled n nodes with k strongly connected components.
Original entry on oeis.org
1, 0, 1, 0, 1, 3, 0, 18, 21, 25, 0, 1606, 1173, 774, 543, 0, 565080, 271790, 122595, 59830, 29281, 0, 734774776, 229224750, 70500705, 25349355, 10110735, 3781503, 0, 3523091615568, 685793359804, 138122171880, 35130437825, 11002159455, 3767987307, 1138779265
Offset: 0
Triangle begins:
1;
0, 1;
0, 1, 3;
0, 18, 21, 25;
0, 1606, 1173, 774, 543;
0, 565080, 271790, 122595, 59830, 29281;
0, 734774776, 229224750, 70500705, 25349355, 10110735, 3781503;
...
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Z(p, f)={my(n=serprec(p, x)); serconvol(p, sum(k=0, n-1, x^k*f(k), O(x^n)))}
G(e, p)={Z(p, k->1/e^(k*(k-1)/2))}
U(e, p)={Z(p, k->e^(k*(k-1)/2))}
DigraphEgf(n, e)={sum(k=0, n, e^(k*(k-1))*x^k/k!, O(x*x^n) )}
T(n)={my(e=2); [Vecrev(p) | p<-Vec(serlaplace(U(e, 1/G(e, exp(y*log(U(e, 1/G(e, DigraphEgf(n, e)))))))))]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) }
A186081
Number of binary relations R on {1,2,...,n} such that the transitive closure of R is the trivial relation.
Original entry on oeis.org
1, 1, 4, 144, 25696, 18082560, 47025585664, 450955726792704, 16260917603754029056, 2253010420928564535951360, 1219004114245442237742488879104, 2601909995433633381004133738019815424, 22040854392120341022554569447470527813779456
Offset: 0
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.
-
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 *)
A365325
Triangular array read by rows. T(n,k) is the number of labeled digraphs (with self loops allowed) on [n] containing exactly k primitive components, n>=0, 0<=k<=n.
Original entry on oeis.org
1, 1, 1, 4, 9, 3, 51, 298, 138, 25, 1831, 40815, 17853, 4494, 543, 166930, 23752151, 7418420, 1861755, 325895, 29281, 36681301, 55427713806, 10701675348, 2105585760, 391017795, 53021223, 3781503
Offset: 0
Triangle begins
1;
1, 1;
4, 9, 3;
51, 298, 138, 25;
1831, 40815, 17853, 4494, 543;
...
-
nn = 6; B[n_] := 2^Binomial[n, 2] n!; 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}]]; primitive =
Select[Import["https://oeis.org/A070322/b070322.txt", "Table"],
Length@# == 2 &][[All, 2]]; 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}];
Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[1/ggf[Exp[- (y (pr[x] - 1) + s[2 x] - (pr[x] - 1))]], {x,
0, nn}], {x, y}]] // Grid
Showing 1-3 of 3 results.
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