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A381102 Irregular triangle read by rows. For each j, 1<=j<=n properly color the vertices of a labeled graph on [n] using each of the first j colors in the color set {c1 Orient the edges according to the strict order on the colors. T(n,k) is the number of such directed graphs containing k descents, n>=0, 0<=k<=binomial(n,2).

%I A381102 #12 Feb 17 2025 03:25:17
%S A381102 1,1,4,1,36,27,9,1,696,983,731,330,93,15,1,27808,60615,72662,59113,
%T A381102 35197,15731,5269,1287,216,22,1,2257888,6803655,11412586,13504721,
%U A381102 12316799,9026017,5427090,2700863,1112555,376459,103002,22203,3619,417,30,1
%N A381102 Irregular triangle read by rows.  For each j, 1<=j<=n properly color the vertices of a labeled graph on [n] using each of the first j colors in the color set {c1<c2<...<cn}.  Orient the edges according to the strict order on the colors. T(n,k) is the number of such directed graphs containing k descents, n>=0, 0<=k<=binomial(n,2).
%C A381102 A descent in a labeled directed graph is an edge s->t such that s>t.
%C A381102 T(n,0) = A289545(n).
%H A381102 Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, <a href="https://arxiv.org/abs/1909.01550">Counting acyclic and strong digraphs by descents</a>, arXiv:1909.01550 [math.CO], 20 Mar 2020.
%e A381102      1;
%e A381102      1;
%e A381102      4,     1;
%e A381102     36,    27,     9,     1;
%e A381102    696,   983,   731,   330,    93,    15,    1;
%e A381102  27808, 60615, 72662, 59113, 35197, 15731, 5269, 1287, 216, 22, 1;
%e A381102  ...
%t A381102 nn = 5; B[n_] :=FunctionExpand[QFactorial[n, (1 + u y)/(1 + y)]] (1 + y)^Binomial[n, 2]; e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, u] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[1/(1 - (e[z] - 1)), {z, 0, nn}], z] /. y -> 1] // Grid
%Y A381102 Cf. A334282(row sums), A289545, A381058, A381192.
%K A381102 nonn,tabf
%O A381102 0,3
%A A381102 _Geoffrey Critzer_, Feb 16 2025