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.

1, 2, 11, 5, 10, 154, 540, 581, 272, 49, 122, 3418, 27304, 90277, 150948, 150519, 95088, 37797, 8714, 893, 3346, 142760, 1938178, 12186976, 42696630, 94605036, 145009210, 161845163, 134933733, 84656743, 39632149, 13481441, 3156845, 455917, 30649

Offset

2,2

Comments

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

Links

Table of n, a(n) for n=2..36.

Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, arXiv:1909.01550 [math.CO], 20 Mar 2020.

Example

Triangle begins:
 1;
 2, 11, 5;
 10, 154, 540, 581, 272, 49;
 122, 3418, 27304, 90277, 150948, 150519, 95088, 37797, 8714, 893;
 3346, 142760, 1938178, 12186976, 42696630, 94605036, 145009210, 161845163, 134933733, 84656743, 39632149, 13481441, 3156845, 455917, 30649;
 ...

Mathematica

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 *)

Crossrefs

Cf. A003030 (row sums), A057273 (another version of the same triangle), A307049, A339590, A005321, A000217.

Sequence in context: A009301 A087552 A124688 * A158616 A127821 A114724

Adjacent sequences: A339804 A339805 A339806 * A339808 A339809 A339810

Keyword

nonn,tabf,changed

Author

Hugo Pfoertner, Dec 28 2020

Extensions

Row 2 added by N. J. A. Sloane, Dec 29 2020

Status

approved