A003454 -id:A003454 - cp's OEIS frontend

A001004 Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.

1, 1, 2, 3, 9, 20, 75, 262, 1117, 4783, 21971, 102249, 489077, 2370142, 11654465, 57916324, 290693391, 1471341341, 7504177738, 38532692207, 199076194985, 1034236705992, 5400337050086, 28329240333758, 149244907249629

Offset: 0

N. J. A. Sloane

Original name: number of symmetric dissections of a polygon.

Also number of 2-connected outerplanar graphs on n unlabeled nodes. - Steven Finch, Dec 09 2004

Cameron, Peter J. Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - N. J. A. Sloane, Apr 18 2014
Guanzhang Hu, Group theory method for enumeration of outerplanar graphs, Acta Math. Appl. Sinica 14 (1998) 381-387.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 0..200
S. R. Finch, Planar graph growth constants.
Steven R. Finch, Planar graph growth constants [Cached copy, with permission of the author]
E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
R. C. Read, On general dissections of a polygon, Preprint (1974)
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.
James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See pp. 7,11.

Cf. A003454, A003455, A003456, A005036, A295260.

f[x_, n_]:=x+Sum[(1/r)*Binomial[s-2, r-1]*Binomial[r+s-1, s]*x^s, {r, 1, n}, {s, 2, n}]; F[x_, n_]:=Series[((3x^2-2*x*f[x, n]+f[x, n]^2)- (2+2*x+7*x^2-4*x*f[x, n]+2*f[x, n]^2)*f[x^2, n]+ 2*f[x^2, n]^2)/(4*(2*f[x^2, n]-1))+Sum[If[Mod[k, d]==0, EulerPhi[d]*f[x^d, n]^(k/d)/k, 0], {k, 3, n}, {d, 1, k}]/2, {x, 0, n}]; F[x, 22] (Finch)

\\ See A295419 for DissectionsModDihedral().
my(v=DissectionsModDihedral(apply(i->1, [1..30])));v[3..#v] \\ Andrew Howroyd, Nov 22 2017

More terms from Esa Peuha (esa.peuha(AT)helsinki.fi), Oct 21 2005

Name clarified by Andrew Howroyd, Nov 22 2017

A003455 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals up to rotation.

Original entry on oeis.org

1, 2, 3, 11, 29, 122, 479, 2113, 9369, 43392, 203595, 975563, 4736005, 23296394, 115811855, 581324861, 2942579633, 15008044522, 77064865555, 398150807179, 2068470765261, 10800665952376, 56658467018647, 298489772155137, 1578702640556193

Offset: 3

N. J. A. Sloane

nonn

Total number of dissections of an n-gon into polygons without reflection. - Sean A. Irvine, May 15 2015

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, A003454, A003456, A005034, A295224, A295495.

\\ See A295495 for DissectionsModCyclic().
DissectionsModCyclic(apply(v->1, [1..30])) \\ Andrew Howroyd, Nov 22 2017

More terms from Sean A. Irvine, May 15 2015

Name clarified by Andrew Howroyd, Nov 22 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

A003456 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation and reflection.

Original entry on oeis.org

1, 2, 5, 17, 62, 275, 1272, 6225, 31075, 158376, 816229, 4251412, 22319056, 117998524, 627573216, 3355499036, 18025442261, 97239773408, 526560862829, 2861189112867, 15595669996482, 85252072993968, 467247847612316, 2567091151780343

Offset: 3

N. J. A. Sloane

nonn

Total number of dissections of an n-gon into polygons with reflection and rooted at a cell. - Sean A. Irvine, May 14 2015

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, A003454, A003455, A005035, A295259.

\\ See A003447 for DissectionsModDihedralRooted()
DissectionsModDihedralRooted(apply(i->1, [1..30]))

More terms from Sean A. Irvine, May 14 2015
Name clarified by Andrew Howroyd, Nov 24 2017
a(15) corrected by Andrew Howroyd, Nov 24 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

A165217 Count of interior bounded regions in a regular 2n-sided polygon dissected by all diagonals parallel to sides.

Original entry on oeis.org

6, 25, 50, 145, 224, 497, 630, 1281, 1606, 2761, 3302, 5265, 5940, 9185, 10472, 14977, 16834, 23161, 25284, 34321, 37720, 49105, 53500, 68225, 73278, 92457, 99470, 122641, 131316, 159681, 169158, 204545, 217210, 258265, 273282, 321937, 338208, 396721, 417380, 483841, 507830

Offset: 3

Chintan (timtamboy63(AT)gmail.com), Sep 08 2009

nonn

The rule is: get a regular polygon with at least 6 sides and an even number of sides (hexagon, octagon, etc.) and pick a point, then pick the point directly clockwise it, draw a line then draw lines parallel to it going through the other points. Then do the same with all the other points. a(n) is the count of interior bounded regions.

Please email me if you can find a pattern!

R. J. Mathar, Tile Count in the Interior of Regular 2n-Gons Dissected by Diagonals Parallel to Sides, arxiv:0911.3434 [math.CO], 2009.
Index to sequences on drawing diagonals in regular polygons

Cf. A003454, A320422

Conjecture: a(2n) = (2*n-1)*(4*n^3-4*n^2+6*n-3)/3. - Thomas Young (tyoung(AT)district16.org), Dec 23 2018

a(6)-a(8) corrected and a(9)-a(10) added by R. J. Mathar, Oct 09 2009
a(11)-a(22) from R. J. Mathar, Nov 19 2009
Typo in a(14) corrected by Thomas Young (tyoung(AT)district16.org), Dec 23 2018
a(23)-a(43) from Christopher Scussel, Jun 25 2023

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A001004 Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.

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A003455 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals up to rotation.

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

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A003456 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation and reflection.

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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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A165217 Count of interior bounded regions in a regular 2n-sided polygon dissected by all diagonals parallel to sides.

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