A382223
Rectangular array read by antidiagonals: T(n,k) is the number of labeled digraphs on [n] along with a (coloring) function c:[n] -> [k] with the property that for all u,v in [n], u->v implies u=0, k>=0.
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 16, 12, 4, 1, 0, 1, 67, 66, 22, 5, 1, 0, 1, 374, 513, 172, 35, 6, 1, 0, 1, 2825, 5769, 1969, 355, 51, 7, 1, 0, 1, 29212, 95706, 33856, 5380, 636, 70, 8, 1, 0, 1, 417199, 2379348, 893188, 125090, 12006, 1036, 92, 9, 1
Offset: 0
Examples
1, 1, 1, 1, 1, 1, 1,... 0, 1, 2, 3, 4, 5, 6,... 0, 1, 5, 12, 22, 35, 51,... 0, 1, 16, 66, 172, 355, 636,... 0, 1, 67, 513, 1969, 5380, 12006,... 0, 1, 374, 5769, 33856, 125090, 352476,...
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.
Programs
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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[e[z]^k, {z, 0, nn}], z], {k, 1, nn}]]], -1];Table[zetapolys /. x -> i, {i, 0, nn}] // Transpose // Grid