A003445 -id:A003445 - cp's OEIS frontend

A003444 Number of dissections of a polygon.

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

nonn

See A220881 for an essentially identical sequence, but with a different offset and a more precise definition. - N. J. A. Sloane, Dec 28 2012

Also number of necklaces of 2 colors with 2n beads and n-2 black ones. - Wouter Meeussen, Aug 03 2002

P. Lisonek, Closed forms for the number of polygon dissections. Journal of Symbolic Computation 20 (1995), 595-601.
R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.
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 = 4..201
D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270 [math.CO], 2012.

Cf. A003445, A003450, A047996, A073020, A220881.

Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#, (n-2)/# ] &)/@Intersection[Divisors[2n], Divisors[n-2]])/(2n), {n, 3, 32}]

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

More terms from Wouter Meeussen, Aug 03 2002

A295633 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 4, 1, 2, 8, 12, 6, 1, 3, 16, 40, 43, 19, 1, 3, 25, 93, 165, 143, 49, 1, 4, 40, 197, 505, 712, 504, 150, 1, 4, 56, 364, 1274, 2548, 2912, 1768, 442, 1, 5, 80, 646, 2878, 7672, 12400, 11976, 6310, 1424, 1, 5, 105, 1050, 5880, 19992, 42840, 58140, 48450, 22610, 4522

Offset: 3

Andrew Howroyd, Nov 24 2017

			Triangle begins: (n >= 3, k >= 1)
1;
1, 1;
1, 1,  1;
1, 2,  4,   4;
1, 2,  8,  12,    6;
1, 3, 16,  40,   43,   19;
1, 3, 25,  93,  165,  143,   49;
1, 4, 40, 197,  505,  712,  504,  150;
1, 4, 56, 364, 1274, 2548, 2912, 1768, 442;
...

Andrew Howroyd, Table of n, a(n) for n = 3..1277

Row sums are A003455.
Column k=3 is A003451.
Diagonals include A001683, A220881, A003445, A220882.
Cf. A295224, A295495, A295634.

\\ See A295495 for DissectionsModCyclic()
T=DissectionsModCyclic(apply(i->y, [1..12]));
for(n=3, #T, for(k=1, n-2, print1(polcoeff(T[n], k), ", ")); print)

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

Original entry on oeis.org

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

N. J. A. Sloane

nonn

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

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
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 A295634.
Cf. A003449, A295419.
Cf. A003444, A003445, A220881.

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)];

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 *)

\\ 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

See Maple program.

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

Name clarified by Andrew Howroyd, Nov 24 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

A220882 Number of (n - 6)-dissections of an n-gon (equivalently, the number of three-dimensional faces of the (n-3)-dimensional associahedron) modulo the cyclic action.

Original entry on oeis.org

1, 2, 16, 93, 505, 2548, 12400, 58140, 266550, 1198564, 5312032, 23263695, 100910001, 434217000, 1855972096, 7887862224, 33359979546, 140492933100, 589495272736, 2465455090098, 10281760786682, 42768958597992, 177499631598976, 735146520745000, 3039095720959424, 12542491305496152

Offset: 6

N. J. A. Sloane, Dec 28 2012

nonn

Douglas Bowman and Alon Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv preprint arXiv:1209.6270 [math.CO], 2012.

Cf. A003444, A003445, A003450, A220881.

C:=n->binomial(2*n,n)/(n+1);
T4:=proc(n) local t1; global C;
t1 :=  (((n-3)*(n-4)^2*(n-5))/(24*n*(2*n-5)))*C(n-2);
if n mod 2 = 0 then t1:=t1+((n-4)^2/(4*n))*C(n/2-2) fi;
if n mod 3 = 0 then t1:=t1+((n-3)/9)*C(n/3-1) fi;
if n mod 6 = 0 then t1:=t1+C(n/6-1)/3 fi;
t1; end;
[seq(T4(n),n=6..40)];

c = CatalanNumber;
T4[n_] := Module[{t1},
t1 = (((n - 3)*(n - 4)^2*(n - 5))/(24*n*(2*n - 5)))*c[n - 2];
If[Mod[n, 2] == 0, t1 = t1 + ((n - 4)^2/(4*n))*c[n/2 - 2]];
If[Mod[n, 3] == 0, t1 = t1 + ((n - 3)/9)*c[n/3 - 1]];
If[Mod[n, 6] == 0, t1 = t1 + c[n/6 - 1]/3]; t1];
Table[T4[n], {n, 6, 40}] (* Jean-François Alcover, Dec 02 2017, from Maple *)

See Maple code.

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A003444 Number of dissections of a polygon.

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A295633 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation.

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

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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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A220882 Number of (n - 6)-dissections of an n-gon (equivalently, the number of three-dimensional faces of the (n-3)-dimensional associahedron) modulo the cyclic action.

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Maple

Mathematica

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