A003445 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation. (Formerly M1859)

1, 2, 8, 40, 165, 712, 2912, 11976, 48450, 195580, 784504, 3139396, 12526605, 49902440, 198499200, 788795924, 3131945190, 12428258796, 49295766000, 195464345440, 774857314042, 3071175790232, 12171403236288, 48233597481200, 191138095393700, 757436171945952

Offset

5,2

Comments

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

Equivalently, the number of two-dimensional faces of the (n-3)-dimensional associahedron modulo the cyclic 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

T. D. Noe, 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);
T31:=proc(n) local t1; global C;
t1 :=  (n-3)^2*(n-4)*C(n-2)/(4*n*(2*n-5));
if n mod 5 = 0 then t1:=t1+(4/5)*C(n/5-1) fi;
if n mod 2 = 0 then t1:=t1+(n-4)*C(n/2-1)/8 fi;
t1; end;
[seq(T31(n), n=5..40)];

Mathematica

Table[t1 = (n - 3)^2*(n - 4)*CatalanNumber[n - 2]/(4*n*(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (4/5)*CatalanNumber[n/5 - 1]]; If[Mod[n, 2] == 0, t1 = t1 + (n - 4)*CatalanNumber[n/2 - 1]/8]; t1, {n, 5, 20}] (* T. D. Noe, Jan 03 2013 *)

Prog

(PARI) \\ See A295495 for DissectionsModCyclic()
{ my(v=DissectionsModCyclic(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 A295633.

Cf. A003444, A003450, A220881.

Cf. A003443, A003448, A003450.

Sequence in context: A127919 A366478 A074092 * A181326 A220964 A231125

Adjacent sequences: A003442 A003443 A003444 * A003446 A003447 A003448

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 25 2017

Status

approved