A382363 Rectangular array read by antidiagonals, T(n,k) is the number of labeled digraphs on [n] along with a (coloring) function c:[n] -> [k] such that for all u,v in [n], u->v implies u<=v and c(u)<=c(v), n>=0, k>=0.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 8, 7, 3, 1, 0, 64, 44, 15, 4, 1, 0, 1024, 508, 129, 26, 5, 1, 0, 32768, 10976, 1962, 284, 40, 6, 1, 0, 2097152, 450496, 54036, 5371, 530, 57, 7, 1, 0, 268435456, 35535872, 2747880, 180424, 11995, 888, 77, 8, 1, 0, 68719476736, 5435551744, 262091808, 10997576, 476165, 23409, 1379, 100, 9, 1

Offset: 0

Author

Geoffrey Critzer, Mar 23 2025

Keywords

• nonn
• tabl

Examples

			  1,    1,     1,     1,      1,      1,       1,...
  0,    1,     2,     3,      4,      5,       6,...
  0,    2,     7,    15,     26,     40,      57,...
  0,    8,    44,   129,    284,    530,     888,...
  0,   64,   508,  1962,   5371,  11995,   23409,...
  0, 1024, 10976, 54036, 180424, 476165, 1072854,...
		

Links

• Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, Discrete Mathematics, Vol. 343, No. 11 (2020), 112041; arXiv preprint, arXiv:1909.01550 [math.CO], 2019-2020; See Table 2.
• R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.

Crossrefs

Cf. A382223, A006125 (column k=1).

Programs

Mathematica

nn = 6; B[n_] := QFactorial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}]; zetapolys = Drop[Map[Expand[InterpolatingPolynomial[#, x]] &,Transpose[Table[Table[B[n], {n, 0, nn}] CoefficientList[Series[1/e[-z]^k, {z, 0, nn}], z], {k,1,nn}]]],-1];Table[zetapolys /. x -> i, {i, 0, nn}] // Transpose // Grid

Formula

Sum_{n>=0} T(n,k)/A005329(n) = 1/e(-x)^k, where e(x) = Sum_{n>=0}x^n/A005329(n).