A003443 -id:A003443 - cp's OEIS frontend

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

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

N. J. A. Sloane

nonn

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

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

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.

A diagonal of A295633.
Cf. A003444, A003450, A220881.
Cf. A003443, A003448, A003450.

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)];

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 *)

\\ 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

See Maple program.

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

Name clarified by Andrew Howroyd, Nov 25 2017

A003442 Number of nonequivalent dissections of an n-gon into (n-3) polygons by nonintersecting diagonals rooted at a cell up to rotation.

Original entry on oeis.org

1, 2, 11, 48, 208, 858, 3507, 14144, 56698, 226100, 898942, 3565920, 14124496, 55887930, 220985795, 873396480, 3450940830, 13633173180, 53855628554, 212750148000, 840496068160, 3320817060132, 13122294166126, 51860761615488

Offset: 4

N. J. A. Sloane

nonn

Number of dissections of regular n-gon into n-3 polygons without reflection and rooted at a cell. - Sean A. Irvine, May 05 2015

The conditions imposed mean that the dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 23 2017

			Case n=5: A pentagon can be dissected into 1 quadrilateral and 1 triangle. Either one of these can be chosen as the root cell so a(n)=2. - _Andrew Howroyd_, Nov 23 2017

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

Andrew Howroyd, Table of n, a(n) for n = 4..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.
Andrey Zabolotskiy, Illustration for n = 4,5,6

Cf. A003443, A003454, A220881, A295622.

DissectionsModCyclicRooted(v)={my(n=#v);
my(q=vector(n)); q[1]=serreverse(x-sum(i=3,#v,x^i*v[i])/x + O(x*x^n));
for(i=2, n, q[i]=q[i-1]*q[1]);
my(vars=variables(q[1]));
my(u(m,r)=substvec(q[r]+O(x^(n\m+1)), vars, apply(t->t^m,vars)));
my(p=O(x*x^n) + sum(i=3, #v, my(c=v[i]); if(c, c*sumdiv(i, d, eulerphi(d)*u(d,i/d))/i)));
vector(n,i,polcoeff(p,i))}
{ my(v=DissectionsModCyclicRooted(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 22 2017

More terms from Sean A. Irvine, May 05 2015

Name clarified by Andrew Howroyd, Nov 22 2017

cp's OEIS Frontend

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

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