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A361242
Number of nonequivalent noncrossing cacti with n nodes up to rotation.
7
1, 1, 1, 2, 7, 26, 144, 800, 4995, 32176, 215914, 1486270, 10471534, 75137664, 547756650, 4047212142, 30255934851, 228513227318, 1741572167716, 13380306774014, 103542814440878, 806476983310180, 6318519422577854, 49769050291536486, 393933908000862866
OFFSET
0,4
COMMENTS
A noncrossing cactus is a connected noncrossing graph (A007297) that is a cactus graph (a tree of edges and polygons).
Since every cactus is an outerplanar graph, every cactus has at least one drawing as a noncrossing graph.
EXAMPLE
The a(3) = 2 nonequivalent cacti have the following blocks:
{{1,2}, {1,3}},
{{1,2,3}}.
Graphically these can be represented:
1 1
/ \ / \
2 3 2----3
.
The a(4) = 7 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}, {2,4}, {3,4}},
{{1,2}, {1,3,4}},
{{1,2}, {2,3,4}},
{{1,2,3,4}}.
Graphically these can be represented:
1---4 1 4 1---4 1 4
| \ | \ | | | / |
2 3 2 3 2---3 2 3
.
1---4 1 4 1---4
| \ | | / | | |
2 3 2---3 2---3
PROG
(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 + 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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Mar 07 2023
STATUS
approved