A001004 Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.
1, 1, 2, 3, 9, 20, 75, 262, 1117, 4783, 21971, 102249, 489077, 2370142, 11654465, 57916324, 290693391, 1471341341, 7504177738, 38532692207, 199076194985, 1034236705992, 5400337050086, 28329240333758, 149244907249629
Offset: 0
References
- Cameron, Peter J. Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - N. J. A. Sloane, Apr 18 2014
- Guanzhang Hu, Group theory method for enumeration of outerplanar graphs, Acta Math. Appl. Sinica 14 (1998) 381-387.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- 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 = 0..200
- S. R. Finch, Planar graph growth constants.
- Steven R. Finch, Planar graph growth constants [Cached copy, with permission of the author]
- E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.
- P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
- T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
- T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
- R. C. Read, On general dissections of a polygon, Preprint (1974)
- Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.
- James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See pp. 7,11.
Programs
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Mathematica
f[x_, n_]:=x+Sum[(1/r)*Binomial[s-2, r-1]*Binomial[r+s-1, s]*x^s, {r, 1, n}, {s, 2, n}]; F[x_, n_]:=Series[((3x^2-2*x*f[x, n]+f[x, n]^2)- (2+2*x+7*x^2-4*x*f[x, n]+2*f[x, n]^2)*f[x^2, n]+ 2*f[x^2, n]^2)/(4*(2*f[x^2, n]-1))+Sum[If[Mod[k, d]==0, EulerPhi[d]*f[x^d, n]^(k/d)/k, 0], {k, 3, n}, {d, 1, k}]/2, {x, 0, n}]; F[x, 22] (Finch) -
PARI
\\ See A295419 for DissectionsModDihedral(). my(v=DissectionsModDihedral(apply(i->1, [1..30])));v[3..#v] \\ Andrew Howroyd, Nov 22 2017
Extensions
More terms from Esa Peuha (esa.peuha(AT)helsinki.fi), Oct 21 2005
Name clarified by Andrew Howroyd, Nov 22 2017
Comments