proposed

approved

editing

proposed

nonn,more,tabf,tabl,changed

proposed

editing

editing

proposed

E. de Panafieu and S. Dovgal, <a href="https://arxiv.org/abs/1903.09454">Symbolic method and directed graph enumeration</a>, arXiv:1903.09454 [math.CO], 2019.

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[-(x + s[x, y])]], {x, 0, nn}], {x, y}]]

Sum_{n>=1} Sum_{k=1..n} T(n,k)*y^k*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(- (x + 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.

allocated for Geoffrey CritzerTriangular array read by rows. T(n,k) is the number of binary relations R on [n] such that the unique idempotent in {R^i:i>=1} contains exactly k non-arcless strongly connected components, n>=0, 0<=k<=n.

1, 1, 1, 3, 9, 4, 25, 277, 162, 48, 543, 38409, 18136, 6912, 1536, 29281, 23169481, 7195590, 2346000, 691200, 122880

0,4

Triangle begins ...

1;

1, 1;

3, 9, 4;

25, 277, 162, 48;

543, 38409, 18136, 6912, 1536;

29281, 23169481, 7195590, 2346000, 691200, 122880;

...

Cf. A002416 (row sums), A003024 (column k=0), A011266 (main diagonal), A370385.

allocated

nonn,more,tabf

Geoffrey Critzer, Feb 19 2024

approved

editing

allocated for Geoffrey Critzer

recycled

allocated

editing

approved

allocated for Tomasz Dziekanski

allocated

recycled

approved

editing