A001004 -id:A001004 - cp's OEIS frontend

A290646 Number of dissections of an n-gon into 3- and 4-gons counted up to rotations and reflections.

1, 2, 2, 7, 14, 53, 171, 691, 2738, 11720, 50486, 224012, 1005468, 4581815, 21093190, 98093226, 459986674, 2173599817, 10340539744, 49496519950, 238240366274, 1152543685463, 5601603835982, 27341242042238, 133977037982121, 658902522544060, 3251446102879398

Offset: 3

Evgeniy Krasko, Sep 03 2017

nonn

			For a(5) = 2 the dissections of a pentagon are: a dissection into 3 triangles; a dissection into one triangle and one quadrangle.

Andrew Howroyd, Table of n, a(n) for n = 3..200
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.
Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.

Cf. A001004 (counted distinctly).

Cf. A001002, A290571, A295260, A295419.

(* See A295419 for DissectionsModDihedral. *)
DissectionsModDihedral[Boole[# == 3 || # == 4]& /@ Range[1, 30]] (* Jean-François Alcover, Sep 25 2019, after Andrew Howroyd *)

\\ See A295419 for DissectionsModDihedral.
DissectionsModDihedral(apply(v->v==3||v==4, [1..25])) \\ Andrew Howroyd, Nov 22 2017

Terms a(16) and beyond from Andrew Howroyd, Nov 22 2017

A290722 Number of dissections of a 2n-gon into polygons with even number of sides counted up to rotations and reflections.

Original entry on oeis.org

1, 2, 4, 13, 48, 238, 1325, 8297, 54519, 373363, 2621872, 18797682, 136969519, 1011903735, 7564219361, 57129086391, 435394899361, 3345082819597, 25885718422329, 201619294539406, 1579629974876090, 12442262963919863, 98483477967355109, 783017782731507416

Offset: 2

Evgeniy Krasko, Sep 03 2017

nonn

			For a(4) = 4 the dissections of an octagon are: two dissections into 3 quadrangles; a dissection into one hexagon and one quadrangle; a dissection into one octagon.

Andrew Howroyd, Table of n, a(n) for n = 2..200
Evgeniy Krasko, Alexander Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.

Cf. A003168 (counted distinctly).

Cf. A001004, A290816, A295419.

\\ See A295419 for DissectionsModDihedral().
select(v->v>0, DissectionsModDihedral(apply(v->v%2==0, [1..50]))) \\ Andrew Howroyd, Nov 22 2017

Terms a(8) and beyond from Andrew Howroyd, Nov 22 2017

A380362 Triangle read by rows: T(n,k) is the number of Halin graphs on n unlabeled nodes with circuit rank k, 3 <= k <= n-1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 3, 2, 1, 0, 0, 0, 3, 6, 3, 1, 0, 0, 0, 0, 7, 11, 3, 1, 0, 0, 0, 0, 4, 24, 17, 4, 1, 0, 0, 0, 0, 0, 24, 51, 26, 4, 1, 0, 0, 0, 0, 0, 12, 89, 109, 36, 5, 1, 0, 0, 0, 0, 0, 0, 74, 265, 194, 50, 5, 1, 0, 0, 0, 0, 0, 0, 27, 371, 660, 345, 65, 6, 1

Offset: 4

Andrew Howroyd, Jan 25 2025

The circuit rank is equal to the number of leaves on the tree before it is extended into a Halin graph by joining up the leaves.
The main diagonal of the graph corresponds with the wheel graphs which have the greatest circuit rank of all Halin graphs.
T(n,k) is also the number of nonequivalent dissections of a k-gon into n-k polygons by nonintersecting diagonals up to rotations and reflections.

			Triangle begins:
  n\k| 3  4  5  6  7   8   9   10  11  12  13
-----+----------------------------------------
   4 | 1;
   5 | 0, 1;
   6 | 0, 1, 1;
   7 | 0, 0, 1, 1;
   8 | 0, 0, 1, 2, 1;
   9 | 0, 0, 0, 3, 2,  1;
  10 | 0, 0, 0, 3, 6,  3,  1;
  11 | 0, 0, 0, 0, 7, 11,  3,   1;
  12 | 0, 0, 0, 0, 4, 24, 17,   4,  1;
  13 | 0, 0, 0, 0, 0, 24, 51,  26,  4,  1;
  14 | 0, 0, 0, 0, 0, 12, 89, 109, 36,  5,  1;
   ...

Andrew Howroyd, Table of n, a(n) for n = 4..1278 (first 50 rows)
Andrew Howroyd, Formulas and PARI Program, Jan 2025.
Eric Weisstein's World of Mathematics, Halin Graph.
Wikipedia, Circuit rank.
Wikipedia, Halin graph.

Row sums are A346779.
Column sums are A001004.
Main diagonal is A000012.
Central coefficients are A000207.
Cf. A295634, A380361.

\\ See PARI Link for program code.
{ my(T=A380361rows(12)); for(i=1, #T, print(T[i])) }

T(n,k) = A295634(k, n-k).

A111563 Number of connected outerplanar graphs on n unlabeled nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 46, 172, 777, 3783, 20074, 111604, 646409, 3846640, 23410035, 144965988, 910898943, 5794179218, 37248630398, 241676806702, 1580880366039, 10416314047854, 69080674190341, 460841447382976, 3090747326749823

Offset: 1

Stefan Vigerske, Nov 17 2005

nonn

Manuel Bodirsky, Éric Fusy, Mihyun Kang and Stefan Vigerske, Enumeration of Unlabeled Outerplanar Graphs, arXiv:math/0511422 [math.CO], 2005-2006.
S. Vigerske, Asymptotic enumeration of unlabeled outerplanar graphs, Diploma thesis, Humboldt University Berlin, 2005.
S. Vigerske, Asymptotic enumeration of unlabeled outerplanar graphs, Slides, Diploma thesis, Humboldt University Berlin, 2005.
Stefan Vigerske, Homepage

Cf. A001004, A097998, A098000.

Generating function and cycle index sum known, see Vigerske or Bodirsky, Fusy, Kang and Vigerske.

A111564 Number of outerplanar graphs on n unlabeled nodes.

Original entry on oeis.org

1, 2, 4, 10, 25, 80, 277, 1150, 5291, 26918, 145744, 828856, 4872771, 29395784, 180857382, 1130700488, 7163245811, 45895629266, 296937363511, 1937625709854, 12739784808937, 84331837321404, 561647630439975, 3761221057579892

Offset: 1

Stefan Vigerske, Nov 17 2005

nonn

M. Bordirsky, É. Fusy, M. Kang and S. Vigerske, Enumeration of Unlabeled Outerplanar Graphs, 2005
S. Vigerske, Asymptotic enumeration of unlabeled outerplanar graphs, Diploma thesis, Humboldt University Berlin, 2005
S. Vigerske, Homepage

Cf. A001004, A097999, A097998, A098000.

Generating function and cycle index sum known, see Vigerske, or Bodirsky, Fusy, Kang and Vigerske.

A290571 Number of dissections of an n-gon into 3- and 5-gons counted up to rotations and reflections.

Original entry on oeis.org

1, 1, 2, 4, 7, 22, 60, 208, 695, 2566, 9451, 36158, 139574, 548347, 2174801, 8719651, 35244472, 143581782, 588858667, 2430036786, 10083626092, 42055927173, 176217259551, 741517642476, 3132564196880, 13281805256068, 56503895845238, 241135999611542

Offset: 3

Evgeniy Krasko, Sep 03 2017

nonn

			For a(5) = 2 the dissections of a pentagon are: a dissection into 3 triangles; a dissection into one pentagon.

Andrew Howroyd, Table of n, a(n) for n = 3..200
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.

Cf. A001004, A290646, A295260.

\\ See A295419 for DissectionsModDihedral().
DissectionsModDihedral(apply(v->v==3||v==5, [1..25])) \\ Andrew Howroyd, Nov 22 2017

Terms a(16) and beyond from Andrew Howroyd, Nov 22 2017

A290816 Number of dissections of an n-gon into polygons with odd number of sides counted up to rotations and reflections.

Original entry on oeis.org

1, 1, 2, 4, 8, 23, 65, 223, 757, 2824, 10559, 40994, 160734, 641420, 2584587, 10528305, 43237978, 178974779, 745814185, 3127246179, 13185588894, 55878618492, 237905685582, 1017225981255, 4366536472758, 18812074137141, 81320795918871, 352638701880227

Offset: 3

Evgeniy Krasko, Sep 03 2017

nonn

			For a(5) = 2 the dissections of a pentagon are: a dissection into 3 triangles; a dissection into one pentagon.

Andrew Howroyd, Table of n, a(n) for n = 3..200
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.

Cf. A049124 (counted distinctly).

Cf. A001004, A290722, A295419.

(* See A295419 for DissectionsModDihedral *)
DissectionsModDihedral[Mod[#, 2]& /@ Range[1, 31]] (* Jean-François Alcover, Sep 25 2019, after Andrew Howroyd *)

\\ See A295419 for DissectionsModDihedral().
DissectionsModDihedral(apply(v->v%2, [1..25])) \\ Andrew Howroyd, Nov 22 2017

Terms a(16) and beyond from Andrew Howroyd, Nov 22 2017

A003457 a(n) = ceiling(Bernoulli(2n)/(-4n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, -1, 14, -140, 1804, -27413, 487469, -10026347, 236192434, -6317862397, 190439655627, -6425425249652, 241207241774251, -10020155328258126, 458387180159766539, -22989944171828251745, 1259023596072554784855, -75008667460769643668557

Offset: 1

N. J. A. Sloane

sign

F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 130.
Douglas C. Ravenel, Complex cobordism theory for number theorists, Lecture Notes in Mathematics, 1326, Springer-Verlag, Berlin-New York, 1988, pp. 123-133.

T. D. Noe, Table of n, a(n) for n = 1..100
R. C. Read, On general dissections of a polygon, Preprint (1974)
Index entries for sequences related to Bernoulli numbers.

Cf. A003414 (floor instead of ceiling).

Table[Ceiling[BernoulliB[2n]/(-4n)], {n, 24}] (* Alonso del Arte, Jul 11 2012 *)

A111757 Number of bipartite 2-connected outerplanar graphs on n unlabeled nodes.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 4, 0, 13, 0, 48, 0, 238, 0, 1325, 0, 8297, 0, 54519, 0, 373363, 0, 2621872, 0, 18797682, 0, 136969519, 0, 1011903735, 0, 7564219361, 0, 57129086391, 0, 435394899361, 0, 3345082819597, 0, 25885718422329, 0, 201619294539406, 0, 1579629974876090

Offset: 1

Stefan Vigerske, Nov 21 2005

nonn

Also the number of bipartite (unlabeled) dissections of a polygon.

M. Bordirsky, É. Fusy, M. Kang and S. Vigerske, Enumeration of Unlabeled Outerplanar Graphs, 2005
S. Vigerske, Asymptotic enumeration of unlabeled outerplanar graphs, Diploma thesis, Humboldt University Berlin, 2005
S. Vigerske, Homepage

Even bisection gives A290722.

Cf. A001004, A111758, A111759, A295419.

\\ See A295419 for DissectionsModDihedral.
{my(N=50); DissectionsModDihedral(vector(N, n, n%2==0)) + vector(N, n, n==2)} \\ Andrew Howroyd, Feb 12 2021

Generating function and cycle index sum known, see Vigerske.

Terms a(21) and beyond from Andrew Howroyd, Feb 12 2021

A375617 Numbers of facially complete 2-connected planar embeddings.

Original entry on oeis.org

0, 0, 1, 3, 6, 15, 32, 94, 295, 1169, 4870, 22110, 102490, 489479, 2370856, 11655722, 57918613, 290697549, 1471349079, 7504192109, 38532719288, 199076246027, 1034236802988, 5400337234593, 28329240686563, 149244907924935, 789351357094770, 4190055030317638

Offset: 1

Eric W. Weisstein, Aug 21 2024

nonn

Eric W. Weisstein, Table of n, a(n) for n = 1..100
James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See pp. 7,11.
Eric Weisstein's World of Mathematics, Facially Complete Planar Embedding.

prism[n_] := Floor[((n - 3)^2 + 6)/12]
tetrahedral[n_] := prism[n - 1]
bipartite[n_] := prism[n - 2]
wheel[n_] := (Mod[n - 1, 2] + 3) 2^Quotient[n - 1, 2]/4 + DivisorSum[n - 1, EulerPhi[#] 2^((n - 1)/#) &]/(2 (n - 1)) - 3
cycle[n_] := Module[{f, F, x},
  f[x_, m_] := x + Sum[(Binomial[s - 2, r - 1] Binomial[r + s - 1, s] x^s)/r, {r, m}, {s, 2, m}];
  F[x_, m_] := Series[((3 x^2 - 2 x f[x, m] + f[x, m]^2) - (2 + 2 x + 7 x^2 - 4 x f[x, m] + 2 f[x, m]^2) f[x^2, m] + 2 f[x^2, m]^2)/(4 (2 f[x^2, m] - 1)) + Sum[If[Mod[k, d] == 0, (EulerPhi[d] f[x^d, m]^(k/d))/k, 0], {k, 3, m}, {d, k}]/2, {x, 0, m}];
  CoefficientList[F[x, n], x][[-1]]]
a[1] = a[2] = 0;
a[n_] := prism[n] + tetrahedral[n] + bipartite[n] + wheel[n] + cycle[n]
Table[a[n], {n, 20}]

a(n) = A001399(n - 6) + A001399(n - 7) + A001399(n - 8) + (A056342(n - 1) - 1) + A001004(n).

cp's OEIS Frontend

A290646 Number of dissections of an n-gon into 3- and 4-gons counted up to rotations and reflections.

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A290722 Number of dissections of a 2n-gon into polygons with even number of sides counted up to rotations and reflections.

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A380362 Triangle read by rows: T(n,k) is the number of Halin graphs on n unlabeled nodes with circuit rank k, 3 <= k <= n-1.

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A111563 Number of connected outerplanar graphs on n unlabeled nodes.

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A111564 Number of outerplanar graphs on n unlabeled nodes.

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A290571 Number of dissections of an n-gon into 3- and 5-gons counted up to rotations and reflections.

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Programs

PARI

Extensions

A290816 Number of dissections of an n-gon into polygons with odd number of sides counted up to rotations and reflections.

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Mathematica

PARI

Extensions

A003457 a(n) = ceiling(Bernoulli(2n)/(-4n)).

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Mathematica

A111757 Number of bipartite 2-connected outerplanar graphs on n unlabeled nodes.

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