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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.

%I A382363 #29 Mar 24 2025 06:12:39
%S A382363 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,
%T A382363 32768,10976,1962,284,40,6,1,0,2097152,450496,54036,5371,530,57,7,1,0,
%U A382363 268435456,35535872,2747880,180424,11995,888,77,8,1,0,68719476736,5435551744,262091808,10997576,476165,23409,1379,100,9,1
%N 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.
%H A382363 Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, <a href="https://doi.org/10.1016/j.disc.2020.112041">Counting acyclic and strong digraphs by descents</a>, Discrete Mathematics, Vol. 343, No. 11 (2020), 112041; <a href="https://arxiv.org/abs/1909.01550">arXiv preprint</a>, arXiv:1909.01550 [math.CO], 2019-2020; See Table 2.
%H A382363 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/18.pdf">Acyclic orientation of graphs,</a> Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
%F A382363 Sum_{n>=0} T(n,k)/A005329(n) = 1/e(-x)^k, where e(x) = Sum_{n>=0}x^n/A005329(n).
%e A382363   1,    1,     1,     1,      1,      1,       1,...
%e A382363   0,    1,     2,     3,      4,      5,       6,...
%e A382363   0,    2,     7,    15,     26,     40,      57,...
%e A382363   0,    8,    44,   129,    284,    530,     888,...
%e A382363   0,   64,   508,  1962,   5371,  11995,   23409,...
%e A382363   0, 1024, 10976, 54036, 180424, 476165, 1072854,...
%t A382363 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
%Y A382363 Cf. A382223, A006125 (column k=1).
%K A382363 nonn,tabl
%O A382363 0,8
%A A382363 _Geoffrey Critzer_, Mar 23 2025