A309524 Triangle read by rows: T(n,k) is the number of simple connected graphs on n nodes with longest path having k nodes, (1 <= k <= n).

1, 0, 1, 0, 0, 2, 0, 0, 1, 5, 0, 0, 1, 2, 18, 0, 0, 1, 3, 17, 91, 0, 0, 1, 3, 29, 86, 734, 0, 0, 1, 4, 42, 176, 864, 10030, 0, 0, 1, 4, 64, 309, 2032, 10243, 248427

Offset: 1

Author

Andrew Howroyd, Sep 06 2019

Keywords

• nonn
• tabl
• more

Comments

Paths here are subgraphs that are isomorphic to a path graph and are measured by the number of vertices they contain rather than the number of edges. No vertex can appear more than once.

Paths with three vertices exist in all connected graphs with at least three vertices. For n > 3, the star graph is the only graph in which longer paths are not possible.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 5;
  0, 0, 1, 2, 18;
  0, 0, 1, 3, 17,  91;
  0, 0, 1, 3, 29,  86,  734;
  0, 0, 1, 4, 42, 176,  864, 10030;
  0, 0, 1, 4, 64, 309, 2032, 10243, 248427;
  ...
		

Links

• Eric Weisstein's World of Mathematics, Longest Path
• Eric Weisstein's World of Mathematics, Traceable Graph

Crossrefs

Row sums are A001349.

Right diagonal is A057864.

Cf. A325455 (circumference = longest cycle).

Cf. A307457.