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A339807 Irregular triangle read by rows: T(n,k) (n>=2, k>=1) is the number of strong digraphs on n nodes with k descents.

%I A339807 #27 Feb 17 2025 03:22:28
%S A339807 1,2,11,5,10,154,540,581,272,49,122,3418,27304,90277,150948,150519,
%T A339807 95088,37797,8714,893,3346,142760,1938178,12186976,42696630,94605036,
%U A339807 145009210,161845163,134933733,84656743,39632149,13481441,3156845,455917,30649
%N A339807 Irregular triangle read by rows: T(n,k) (n>=2, k>=1) is the number of strong digraphs on n nodes with k descents.
%C A339807 T(n,1) = A005321(n-1). Length of row n = binomial(n,2). It appears that T(n,binomial(n,2)) = A348901(n-1). - _Geoffrey Critzer_, Feb 12 2025
%H A339807 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 A339807 Triangle begins:
%e A339807  1;
%e A339807  2, 11, 5;
%e A339807  10, 154, 540, 581, 272, 49;
%e A339807  122, 3418, 27304, 90277, 150948, 150519, 95088, 37797, 8714, 893;
%e A339807  3346, 142760, 1938178, 12186976, 42696630, 94605036, 145009210, 161845163, 134933733, 84656743, 39632149, 13481441, 3156845, 455917, 30649;
%e A339807  ...
%t A339807 nn = 8; B[n_] := FunctionExpand[QFactorial[n, (1 + u y)/(1 + y)]] (1 + y)^Binomial[n, 2]; g[z_] := Sum[(1 + u y)^Binomial[n, 2] z^n/FunctionExpand[QFactorial[n, (1 + u y)/(1 + y)]], {n, 0, nn}]; egf[eggf_] := Normal[Series[eggf, {z, 0, nn}]] /.Table[z^i -> z^i*B[i]/i!, {i, 1, nn + 1}]; Map[Drop[#, 1] &, Drop[Map[CoefficientList[#, u] &, Table[n!, {n, 0, nn}]CoefficientList[Series[-Log[egf[1/g[z]]], {z, 0, nn}], z] /. y -> 1], 2]] // Grid (* _Geoffrey Critzer_, Feb 12 2025 *)
%Y A339807 Cf. A003030 (row sums), A057273 (another version of the same triangle), A307049, A339590, A005321, A000217.
%K A339807 nonn,tabf
%O A339807 2,2
%A A339807 _Hugo Pfoertner_, Dec 28 2020
%E A339807 Row 2 added by _N. J. A. Sloane_, Dec 29 2020