A003454 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation.

1, 2, 6, 25, 107, 509, 2468, 12258, 61797, 315830, 1630770, 8498303, 44629855, 235974495, 1255105304, 6710883952, 36050676617, 194478962422, 1053120661726, 5722375202661, 31191334491891, 170504130213135, 934495666529380, 5134182220623958, 28270742653671621

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Total number of dissections of an n-gon into polygons without reflection and rooted at a cell. - Sean A. Irvine, May 14 2015

Say two n-gons are equivalent (or in the same convexity class) if there is a bijection between the edges and vertices which preserves inclusion of vertices and edges, preserves the handedness of the polygon (does not reflect the polygon over a line), maps vertices of the convex hulls to each other, and induces an equivalence on each nontrivially connected component of Hull(X) \ X. This sequence is the number of convexity classes for an n-gon, up to rotation. - Griffin N. Macris, Mar 02 2021

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..200
• P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
• Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.

Crossrefs

Cf. A001004, A003455, A003456, A005033, A295222.

Cf. A003442.

Programs

PARI

\\ See A003442 for DissectionsModCyclicRooted.
DissectionsModCyclicRooted(apply(i->1, [1..30])) \\ Andrew Howroyd, Nov 22 2017

Formula

G.f.: -f(x) - (f(x)^2 + f(x^2))/2 + Sum_{k>=1} (phi(k)/k)*log(1/(1 - f(x^k))), where phi(k) is Euler's Totient function and f(x) = (1 + x - sqrt(1 - 6x + x^2))/4 is related to the o.g.f. for A001003. - Griffin N. Macris, Mar 02 2021

Extensions

More terms from Sean A. Irvine, May 14 2015

Name clarified by Andrew Howroyd, Nov 22 2017