A372173 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with exactly k triangles, 0 <= k <= binomial(n,3).

1, 0, 1, 1, 1, 4, 1, 1, 0, 1, 7, 5, 4, 2, 2, 1, 0, 1, 0, 0, 1, 24, 16, 23, 12, 15, 8, 7, 4, 4, 1, 3, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 69, 79, 113, 103, 105, 83, 73, 58, 45, 34, 31, 22, 14, 16, 10, 4, 8, 5, 2, 3, 2, 2, 2, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 0

Author

Gus Wiseman, Apr 23 2024

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1
  0
  1
  1 1
  4 1 1 0 1
  7 5 4 2 2 1 0 1 0 0 1
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..340 (rows 0..10)
• Gus Wiseman, All unlabeled simple graphs covering 5 vertices, grouped by number of triangles.

Crossrefs

Row sums are A002494, labeled A006129.

Row lengths are A050407.

The non-covering version is A263340, labeled A372170.

Counting edges instead of triangles gives A370167, labeled A054548.

The labeled version is A372167.

Column k = 0 is A372169, labeled A372168 (non-covering A213434).

Column k = 1 is A372174, labeled A372171.

Column k = 1 is also the covering case of A372194, labeled A372172.

A000088 counts unlabeled graphs, labeled A006125.

A001858 counts acyclic graphs, unlabeled A005195.

A372176 counts labeled graphs by directed cycles, covering A372175.

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

Extensions

a(21) onwards from Andrew Howroyd, Dec 29 2024