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
A365547
Triangular array read by rows. T(n,k) is the number of convergent Boolean relation matrices on [n] containing exactly k strongly connected components, n>=0, 0<=k<=n.
Original entry on oeis.org
1, 0, 2, 0, 3, 12, 0, 139, 126, 200, 0, 25575, 17517, 9288, 8688, 0, 18077431, 8457840, 3545350, 1435920, 936992, 0, 47024942643, 14452288791, 4277647665, 1422744780, 485315280, 242016192
Offset: 0
Triangle begins ...
1;
0, 2;
0, 3, 12;
0, 139, 126, 200;
0, 25575, 17517, 9288, 8688;
0, 18077431, 8457840, 3545350, 1435920, 936992;
...
- D. A. Gregory, S. Kirkland, and N. J. Pullman, Power convergent Boolean matrices, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.
- G. Markowsky, Bounds on the index and period of a binary relation on a finite set, Semigroup Forum, Vol 13 (1977), 253-259.
- E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
- R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.
- D. Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.
-
nn = 6; B[n_] := n! 2^Binomial[n, 2]; 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}];Table[Take[(Table[B[n], {n, 0, nn}] CoefficientList[Series[1/ggf[Exp[-(y pr[x] - y + y x)]], {x, 0, nn}], {x, y}])[[i]], i], {i, 1, 7}] // Grid
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