A263340 Triangle read by rows: T(n,k) is the number of graphs with n vertices containing k triangles.

1, 1, 2, 3, 1, 7, 2, 1, 0, 1, 14, 7, 5, 2, 3, 1, 0, 1, 0, 0, 1, 38, 23, 28, 14, 18, 9, 7, 5, 4, 1, 4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 107, 102, 141, 117, 123, 92, 80, 63, 49, 35, 35, 23, 15, 17, 10, 4, 9, 5, 2, 3, 3, 2, 2, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 0

Author

Christian Stump, Oct 15 2015

Keywords

• nonn
• tabf

Comments

Row sums give A000088.

First column is A006785.

Row lengths are 1 + binomial(n,3). - Geoffrey Critzer, Apr 13 2017

Examples

			Triangle begins:
  1;
  1;
  2;
  3,1;
  7,2,1,0,1;
  14,7,5,2,3,1,0,1,0,0,1;
  38,23,28,14,18,9,7,5,4,1,4,1,1,1,0,0,1,0,0,0,1;
  ...
		

Links

• Pontus von Brömssen, Rows n = 0..10, flattened
• FindStat - Combinatorial Statistic Finder, The number of triangles of a graph.
• Gus Wiseman, The graphs counted under row n = 5.

Crossrefs

Row sums are A000088, labeled A006125.

Column k = 0 is A006785 (lab A213434), covering A372169 (lab A372168).

Counting edges gives A008406 (lab A084546), covering A370167 (lab A054548).

Row lengths are A050407.

The labeled version is A372170, covering A372167.

The covering case is A372173, sums A002494, labeled A006129.

Column k = 1 is A372194 (lab A372172), covering A372174 (lab A372171).

A001858 counts acyclic graphs, unlabeled A005195.

A372176 counts labeled graphs by directed cycles, covering A372175.

Cf. A000055, A053530, A137917, A137918, A140637, A144958, A322700, A370169.

Programs

Mathematica

Table[Table[Count[Table[Tr[MatrixPower[AdjacencyMatrix[GraphData[{n, i}]], 3]]/6, {i, 1, NumberOfGraphs[n]}], k], {k, 0, Binomial[n, 3]}], {n, 1, 7}] (* Geoffrey Critzer, Apr 13 2017 *)

Extensions

Row 7 from Geoffrey Critzer, Apr 13 2017

T(0,0)=1 prepended by Alois P. Heinz, Apr 13 2017