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).

1, 1, 4, 1, 36, 27, 9, 1, 696, 983, 731, 330, 93, 15, 1, 27808, 60615, 72662, 59113, 35197, 15731, 5269, 1287, 216, 22, 1, 2257888, 6803655, 11412586, 13504721, 12316799, 9026017, 5427090, 2700863, 1112555, 376459, 103002, 22203, 3619, 417, 30, 1

Offset

0,3

Comments

A descent in a labeled directed graph is an edge s->t such that s>t.

T(n,0) = A289545(n).

Links

Table of n, a(n) for n=0..41.

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

     1;
     1;
     4,     1;
    36,    27,     9,     1;
   696,   983,   731,   330,    93,    15,    1;
 27808, 60615, 72662, 59113, 35197, 15731, 5269, 1287, 216, 22, 1;
 ...

Mathematica

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

Crossrefs

Cf. A334282(row sums), A289545, A381058, A381192.

Sequence in context: A011801 A169656 A362589 * A303987 A297900 A363819

Adjacent sequences: A381099 A381100 A381101 * A381103 A381104 A381105

Keyword

nonn,tabf

Author

Geoffrey Critzer, Feb 16 2025

Status

approved