cp's OEIS Frontend

A381192 Irregular triangle read by rows. Properly color the vertices of a simple labeled graph on [n] using exactly n colors c_1 (in other words, use each color exactly once). Orient the edges according to the strict order on the colors. T(n,k) is the number of such graphs with exactly k descents, n>=0, 0<=k<=binomial(n,2).

%I A381192 #14 Feb 24 2025 19:41:40
%S A381192 1,1,3,1,21,19,7,1,315,516,419,208,65,12,1,9765,24186,31445,27488,
%T A381192 17538,8420,3050,816,153,18,1,615195,2080323,3769767,4754751,4592847,
%U A381192 3555479,2257723,1188595,519745,187705,55237,12941,2325,301,25,1
%N A381192 Irregular triangle read by rows.  Properly color the vertices of a simple labeled graph on [n] using exactly n colors c_1<c_2<...<c_n (in other words, use each color exactly once).  Orient the edges according to the strict order on the colors.  T(n,k) is the number of such graphs with exactly k descents, n>=0, 0<=k<=binomial(n,2).
%C A381192 A descent in a labeled directed graph is an edge s->t such that s>t.
%C A381192 T(n,0) = A005329(n).
%C A381192 Sum_{k>=0} T(n,k)*k = A005329(n)*n(n-1)/8.
%H A381192 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 A381192      1;
%e A381192      1;
%e A381192      3,     1;
%e A381192     21,    19,     7,     1;
%e A381192    315,   516,   419,   208,    65,   12,   1;
%e A381192   9765, 24186, 31445, 27488, 17538, 8420, 3050, 816, 153, 18, 1;
%e A381192   ...
%t A381192 nn = 6; 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 - z), {z, 0, nn}], z] /. y -> 1] // Grid
%Y A381192 CF. A005329, A381058, A011266 (row sums), A381102.
%K A381192 nonn,tabf
%O A381192 0,3
%A A381192 _Geoffrey Critzer_, Feb 16 2025