cp's OEIS Frontend

A380336 Triangular array read by rows. T(n,k) is the number of ways to choose a size k subset S of [n] and form a labeled acyclic digraph on S. Then form another labeled acyclic digraph on [n]-S. For each pair u in S and v in [n]-S add the directed edge u->v or not, n>=0, 0<=k<=n.

%I A380336 #16 Jan 23 2025 00:22:52
%S A380336 1,1,1,3,4,3,25,36,36,25,543,800,864,800,543,29281,43440,48000,48000,
%T A380336 43440,29281,3781503,5621952,6255360,6400000,6255360,5621952,3781503,
%U A380336 1138779265,1694113344,1888975872,1946112000,1946112000,1888975872,1694113344,1138779265
%N A380336 Triangular array read by rows.  T(n,k) is the number of ways to choose a size k subset S of [n] and form a labeled acyclic digraph on S.  Then form another labeled acyclic digraph on [n]-S. For each pair u in S and v in [n]-S add the directed edge u->v or not, n>=0, 0<=k<=n.
%H A380336 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.
%H A380336 R. P. Stanley, <a href="https://doi.org/10.1016/j.disc.2006.03.010">Acyclic orientation of graphs</a>, Discrete Math. 5 (1973), 171-178.
%F A380336 Sum_{n>=0} T(n,k)*y^k*x^n/(2^binomial(n,2)*n!) = 1/E(-y*x)*1/E(-x) where E(x) = Sum_{n>=0} x^n/(2^binomial(n,2)*n!).
%F A380336 T(n,k) = binomial(n,k)*A003024(k)*A003024(n-k)*2^(k*(n-k)). - _Alois P. Heinz_, Jan 22 2025
%e A380336 Triangle T(n,k) begins:
%e A380336      1;
%e A380336      1,     1;
%e A380336      3,     4,     3;
%e A380336     25,    36,    36,    25;
%e A380336    543,   800,   864,   800,   543;
%e A380336  29281, 43440, 48000, 48000, 43440, 29281;
%e A380336  ...
%t A380336 nn = 6; B[n_] := n! 2^Binomial[n, 2]; e[z_] := Sum[z^n/B[n], {n, 0, nn}]; Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[1/e[-u z]*1/e[-z], {z, 0, nn}], {z, u}]] // Grid
%Y A380336 Cf. A339934 (row sums), A003024 (column k=0 and main diagonal).
%K A380336 nonn,tabl
%O A380336 0,4
%A A380336 _Geoffrey Critzer_, Jan 21 2025