A003442 -id:A003442 - cp's OEIS frontend

A003454 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation.

1, 2, 6, 25, 107, 509, 2468, 12258, 61797, 315830, 1630770, 8498303, 44629855, 235974495, 1255105304, 6710883952, 36050676617, 194478962422, 1053120661726, 5722375202661, 31191334491891, 170504130213135, 934495666529380, 5134182220623958, 28270742653671621

Offset: 3

N. J. A. Sloane

nonn

Total number of dissections of an n-gon into polygons without reflection and rooted at a cell. - Sean A. Irvine, May 14 2015

Say two n-gons are equivalent (or in the same convexity class) if there is a bijection between the edges and vertices which preserves inclusion of vertices and edges, preserves the handedness of the polygon (does not reflect the polygon over a line), maps vertices of the convex hulls to each other, and induces an equivalence on each nontrivially connected component of Hull(X) \ X. This sequence is the number of convexity classes for an n-gon, up to rotation. - Griffin N. Macris, Mar 02 2021

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 = 3..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.

Cf. A001004, A003455, A003456, A005033, A295222.

Cf. A003442.

\\ See A003442 for DissectionsModCyclicRooted.
DissectionsModCyclicRooted(apply(i->1, [1..30])) \\ Andrew Howroyd, Nov 22 2017

G.f.: -f(x) - (f(x)^2 + f(x^2))/2 + Sum_{k>=1} (phi(k)/k)*log(1/(1 - f(x^k))), where phi(k) is Euler's Totient function and f(x) = (1 + x - sqrt(1 - 6x + x^2))/4 is related to the o.g.f. for A001003. - Griffin N. Macris, Mar 02 2021

More terms from Sean A. Irvine, May 14 2015

Name clarified by Andrew Howroyd, Nov 22 2017

A220881 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation.

Original entry on oeis.org

1, 1, 4, 12, 43, 143, 504, 1768, 6310, 22610, 81752, 297160, 1086601, 3991995, 14732720, 54587280, 202997670, 757398510, 2834510744, 10637507400, 40023636310, 150946230006, 570534578704, 2160865067312, 8199711378716, 31170212479588, 118686578956272

Offset: 4

N. J. A. Sloane, Dec 28 2012

nonn

This is almost identical to A003444, but has a different offset and a more precise definition.
In other words, the number of almost-triangulations of an n-gon modulo the cyclic action.
Equivalently, the number of edges of the (n-3)-dimensional associahedron modulo the cyclic action.
The dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 25 2017
Also number of necklaces of 2 colors with 2n-4 beads and n black ones. - Wouter Meeussen, Aug 03 2002

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

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, A003445.
Cf. A003442, A003447, A003449.

C:=n->binomial(2*n,n)/(n+1);
T2:= proc(n) local t1; global C;
t1 :=  (n-3)*C(n-2)/(2*n);
if n mod 4 = 0 then t1:=t1+C(n/4-1)/2 fi;
if n mod 2 = 0 then t1:=t1+C(n/2-1)/4 fi;
t1; end;
[seq(T2(n),n=4..40)];

c[n_] := Binomial[2*n, n]/(n+1);
T2[n_] := Module[{t1}, t1 = (n-3)*c[n-2]/(2*n); If[Mod[n, 4] == 0, t1 = t1 + c[n/4-1]/2]; If[Mod[n, 2] == 0, t1 = t1 + c[n/2-1]/4]; t1];
Table[T2[n], {n, 4, 40}] (* Jean-François Alcover, Nov 23 2017, translated from Maple *)
a[n_] := Sum[EulerPhi[d]*Binomial[(2n-4)/d, n/d], {d, Divisors[GCD[2n-4, n] ]}]/(2n-4);
Array[a, 30, 4] (* Jean-François Alcover, Dec 02 2017, after Andrew Howroyd *)

a(n) = if(n>=4, sumdiv(gcd(2*n-4, n), d, eulerphi(d)*binomial((2*n-4)/d, n/d))/(2*n-4)) \\ Andrew Howroyd, Nov 25 2017

a(n) = (1/(2n-4)) Sum_{d |(2n-4, n)} phi(d)*binomial((2n-4)/d, n/d) for n >= 4. - Wouter Meeussen, Aug 03 2002

Name clarified by Andrew Howroyd, Nov 25 2017

A003443 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals rooted at a cell up to rotation.

Original entry on oeis.org

1, 3, 24, 150, 825, 4205, 20384, 95472, 436050, 1954150, 8629528, 37665030, 162845865, 698599125, 2977488000, 12620579140, 53243068230, 223707978090, 936619554000, 3909283969500, 16272003594658, 67565854800378, 279942274434624

Offset: 5

N. J. A. Sloane

nonn

Number of dissections of regular n-gon into n-4 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 either 1 pentagon and n-5 triangles or 2 quadrilaterals and n-6 triangles. - 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 = 5..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.

Cf. A003442, A003454.

\\ See A003442 for DissectionsModCyclicRooted()
{ my(v=DissectionsModCyclicRooted(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 22 2017

More terms from Sean A. Irvine, May 05 2015

Name clarified and offset changed by Andrew Howroyd, Nov 22 2017

A295622 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals rooted at a cell up to rotation.

Original entry on oeis.org

3, 11, 24, 46, 75, 117, 168, 236, 315, 415, 528, 666, 819, 1001, 1200, 1432, 1683, 1971, 2280, 2630, 3003, 3421, 3864, 4356, 4875, 5447, 6048, 6706, 7395, 8145, 8928, 9776, 10659, 11611, 12600, 13662, 14763, 15941, 17160, 18460, 19803, 21231, 22704, 24266

Offset: 5

Andrew Howroyd, Nov 24 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 5..500
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.

Cf. A003442, A003451, A003452, A003453.

\\ See A003442 for DissectionsModCyclicRooted()
{ my(v=DissectionsModCyclicRooted(apply(i->y + O(y^4), [1..40]))); apply(p->polcoeff(p, 3), v[5..#v]) }

Conjectures from Colin Barker, Nov 25 2017: (Start)
G.f.: x^5*(3 + 5*x - x^2 - x^3) / ((1 - x)^4*(1 + x)^2).
a(n) = (n-4)*(-5 + (-1)^n - 4*n + 2*n^2) / 8 for n>4.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>10.
(End)
a(n) = Sum_{k=0..n-5} f(k), where f(n) = Sum_{k=0..n} (3 + lcm(k, 2)) (conjecture). - Jon Maiga, Nov 28 2018

cp's OEIS Frontend

A003454 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation.

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A220881 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation.

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A003443 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals rooted at a cell up to rotation.

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Author

Keywords

Comments

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Links

Crossrefs

Programs

PARI

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A295622 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals rooted at a cell up to rotation.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula