A003450 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation and reflection. (Formerly M1673)

1, 2, 6, 24, 89, 371, 1478, 6044, 24302, 98000, 392528, 1570490, 6264309, 24954223, 99253318, 394409402, 1565986466, 6214173156, 24647935156, 97732340680, 387428854374, 1535588541762, 6085702368796, 24116801236744, 95569050564444, 378718095630676

Offset

5,2

Comments

In other words, the number of (n - 5)-dissections of an n-gon modulo the dihedral action.

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

The dissection will always be composed of either 1 pentagon and n-5 triangles or 2 quadrilaterals and n-6 triangles. - Andrew Howroyd, Nov 24 2017

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 = 5..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.

Formula

See Maple program.

Maple

C:=n->binomial(2*n, n)/(n+1);
T32:=proc(n) local t1; global C;
if n mod 2 = 0 then
t1 :=  (n-3)^2*(n-4)*C(n-2)/(8*n*(2*n-5));
if n mod 5 = 0 then t1:=t1+(2/5)*C(n/5-1) fi;
if n mod 2 = 0 then t1:=t1+((3*(n-4)*(n-1))/(16*(n-3)))*C(n/2-1) fi;
else
t1 :=  (n-3)^2*(n-4)*C(n-2)/(8*n*(2*n-5));
if n mod 5 = 0 then t1:=t1+(2/5)*C(n/5-1) fi;
if n mod 2 = 1 then t1:=t1+((n^2-2*n-11)/(8*(n-4)))*C((n-3)/2) fi;
fi;
t1; end;
[seq(T32(n), n=5..40)];

Mathematica

c = CatalanNumber;
T32[n_] := Module[{t1}, If[EvenQ[n], t1 = (n-3)^2*(n-4)*c[n-2]/(8*n*(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (2/5)*c[n/5-1]]; If[EvenQ[n], t1 = t1 + ((3*(n-4)*(n-1))/(16*(n-3)))*c[n/2-1]], t1 = (n-3)^2*(n-4)*c[n-2]/(8*n *(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (2/5) * c[n/5-1]]; If[OddQ[n], t1 = t1 + ((n^2 - 2*n - 11)/(8*(n-4)))*c[(n-3)/2]]]; t1];
Table[T32[n], {n, 5, 40}] (* Jean-François Alcover, Dec 11 2017, translated from Maple *)

Prog

(PARI) \\ See A295419 for DissectionsModDihedral()
{ my(v=DissectionsModDihedral(apply(i->if(i>=3&&i<=5, y^(i-3) + O(y^3)), [1..30]))); apply(p->polcoeff(p, 2), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017

Crossrefs

A diagonal of A295634.

Cf. A003449, A295419.

Cf. A003444, A003445, A220881.

Sequence in context: A217527 A293774 A226037 * A192466 A367274 A361752

Adjacent sequences: A003447 A003448 A003449 * A003451 A003452 A003453

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Name clarified by Andrew Howroyd, Nov 24 2017

Status

approved