A370385 Triangular array read by rows. T(n,k) is the number of binary relations R on [n] such that the unique idempotent relation in {R^i:i>=1} is a quasi-order containing exactly k strongly connected components.
1, 1, 3, 4, 139, 66, 48, 25575, 9280, 3072, 1536, 18077431, 4498530, 1174800, 322560, 122880
Offset: 1
Examples
Triangle begins:
1;
1;
3, 4;
139, 66, 48;
25575, 9280, 3072, 1536;
18077431, 4498530, 1174800, 322560, 122880;
...
Links
- E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
Crossrefs
Programs
-
Mathematica
nn = 5; B[n_] := n! 2^Binomial[n, 2]; s[x_, y_] := y x + (3 y + y^2) x^2/2! + (139 y + 3 y^2 + 2 y^3) x^3/3! + (25575 y + 103 y^2 + 12 y^3 + 6 y^4) x^4/ 4! + (18077431 y + 4815 y^2 + 230 y^3 + 60 y^4 + 24 y^5) x^5/5! ; 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[-s[x, y]]], {x, 0, nn}], {x, y}]]
Formula
Sum_{n>=1} Sum_{k=1..n} T(n,k)*y^k*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(-s(x,y))) where E(x) = Sum_{n>=0} x^n/(n!*2^binomial(n,2)) and @ is the exponential Hadamard product (see Panafieu and Dovgal) and s(x,y) is the e.g.f. for A367948.