A003449 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation and reflection.

1, 1, 3, 7, 24, 74, 259, 891, 3176, 11326, 40942, 148646, 543515, 1996212, 7367075, 27294355, 101501266, 378701686, 1417263770, 5318762098, 20011847548, 75473144396, 285267393358, 1080432637662, 4099856060808, 15585106611244, 59343290815356

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn

Comments

In other words, the number of almost-triangulations of an n-gon modulo the dihedral action.

Equivalently, the number of edges of the (n-3)-dimensional associahedron modulo the dihedral action.

The dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 24 2017

See Theorem 30 of Bowman and Regev (although there appears to be a typo in the formula - see Maple code below). - N. J. A. Sloane, Dec 28 2012

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 = 4..200
• D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270 [math.CO], 2012.
• 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

A diagonal of A295634.

Cf. A003450, A295419.

Programs

Maple

C:=n->binomial(2*n,n)/(n+1);
T30:=proc(n) local t1; global C;
if n mod 2 = 0 then
t1:=(1/4-(3/(4*n)))*C(n-2) + (3/8)*C(n/2-1) + (1-3/n)*C(n/2-2);
if n mod 4 = 0 then t1:=t1+C(n/4-1)/4 fi;
else
t1:=(1/4-(3/(4*n)))*C(n-2) + (1/2)*C((n-3)/2);
fi;
t1; end;
[seq(T30(n),n=4..40)]; # N. J. A. Sloane, Dec 28 2012

Mathematica

c = CatalanNumber;
T30[n_] := Module[{t1}, If[EvenQ[n], t1 = (1/4 - (3/(4*n)))*c[n - 2] + (3/8)*c[n/2 - 1] + (1 - 3/n)*c[n/2 - 2]; If[Mod[n, 4] == 0, t1 = t1 + c[n/4 - 1]/4], t1 = (1/4 - (3/(4*n)))*c[n-2] + (1/2)*c[(n-3)/2]]; t1];
Table[T30[n], {n, 4, 40}] (* Jean-François Alcover, Dec 14 2017, after N. J. A. Sloane *)

PARI

\\ See A295419 for DissectionsModDihedral()
{ my(v=DissectionsModDihedral(apply(i->if(i>=3&&i<=4, y^(i-3) + O(y^2)), [1..25]))); apply(p->polcoeff(p, 1), v[4..#v]) } \\ Andrew Howroyd, Nov 24 2017

Extensions

Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012

Name clarified by Andrew Howroyd, Nov 24 2017