A361243 Number of nonequivalent noncrossing cacti with n nodes up to rotation and reflection.

1, 1, 1, 2, 5, 17, 79, 421, 2537, 16214, 108204, 743953, 5237414, 37574426, 273889801, 2023645764, 15128049989, 114256903169, 870786692493, 6690155544157, 51771411793812, 403238508004050, 3159259746188665, 24884525271410389, 196966954270163612

Offset: 0

Author

Andrew Howroyd, Mar 07 2023

Keywords

• nonn

Comments

A noncrossing cactus is a connected noncrossing graph (A007297) that is a cactus graph (a tree of edges and polygons).

Examples

			The a(4) = 5 nonequivalent cacti have the following blocks:
  {{1,2}, {1,3}, {1,4}},
  {{1,2}, {1,3}, {3,4}},
  {{1,2}, {1,4}, {2,3}},
  {{1,2}, {1,3,4}},
  {{1,2,3,4}}.
Graphically these can be represented:
   1---4    1   4    1---4    1---4    1---4
   | \      | \ |    |        | \ |    |   |
   2   3    2   3    2---3    2   3    2---3
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Wikipedia, Cactus graph.
• Index entries for sequences related to cacti.

Crossrefs

Cf. A003168, A007297, A361239, A361242.

Programs

PARI

\\ Here F(n) is the g.f. of A003168.
F(n) = {1 + serreverse(x/((1+2*x)*(1+x)^2) + O(x*x^n))}
seq(n) = {my(f=F(n-1)); Vec(1/(1 - x*subst(f + O(x^(n\2+1)), x, x^2)) + 1 + intformal(f) - sum(d=2, n, eulerphi(d) * log(1-subst(x*f^2 + O(x^(n\d+1)),x,x^d)) / d), -n-1)/2}