A113201 -id:A113201 - cp's OEIS frontend

A078666 Number of isomorphism classes of simple quadrangulations of the sphere having n+2 vertices and n faces, minimal degree 3, with orientation-reversing isomorphisms permitted.

1, 0, 1, 1, 3, 3, 12, 19, 64, 155, 510, 1514, 5146, 16966, 58782, 203269, 716607, 2536201, 9062402, 32533568, 117498072, 426212952, 1553048548, 5681011890, 20858998805, 76850220654, 284057538480, 1053134292253, 3915683667721

Offset: 6

Slavik V. Jablan and Brendan McKay Feb 06 2003

nonn

Number of basic polyhedra with n vertices.
Initial terms of sequence coincide with A007022. Starting from n=12, to it is added the number of simple 4-regular 4-edge-connected but not 3-connected plane graphs on n nodes (A078672). As a result we obtain the number of basic polyhedra.
a(n) counts 4-valent 4-edge-connected planar maps (or plane graphs on a sphere) up to reflection with no regions bounded by just 2 edges. Conway called such maps "basic polyhedra" and used them in his knot notation. 2-edge-connected maps (which start occurring from n=12) are not taken into account here because they generate only composite knots and links. - Andrey Zabolotskiy, Sep 18 2017

			G.f. = x^6 + x^8 + x^9 + 3*x^10 + 3*x^11 + 12*x^12 + 19*x^13 + 64*x^14 + ...
From _Allan Bickle_, May 13 2024: (Start)
For n=6, the unique graph is the octahedron.
For n=8, the unique graph is the square of an 8-cycle.
For n=9, the unique graph is the dual of the Herschel graph. (End)

J. H. Conway, An enumeration of knots and links and some of their related properties. Computational Problems in Abstract Algebra, Proc. Conf. Oxford 1967 (Ed. J. Leech), 329-358. New York: Pergamon Press, 1970.

G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
A. Caudron, Classification des noeuds et des enlacements, Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982.
Alain Caudron, Classification des noeuds et des enlacements (Thèse et additifs), Univ. Paris-Sud, 1989 [Scanned copy, included with permission]. Contains additional material.
CombOS - Combinatorial Object Server, generate planar graphs
S. V. Jablan, Ordering Knots
S. V. Jablan, L. M. Radović, and R. Sazdanović, Basic polyhedra in knot theory Kragujevac J. Math., 28 (2005), 155-164.
The Knot Atlas, Conway Notation.
Index entries for sequences related to knots

Cf. A007022, A078672, A113201, A072552, A111361.

Cf. A292515 (abstract planar graphs with same restrictions).

Name and offset corrected by Andrey Zabolotskiy, Aug 22 2017

A002880 Number of 3-connected nets with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 9, 11, 37, 79, 249, 671, 2182, 6692, 22131, 72405, 243806, 822788, 2815119, 9679205, 33551192, 116900081, 409675567, 1442454215, 5102542680, 18124571838, 64634480340, 231334873091, 830828150081, 2993489821771

Offset: 6

N. J. A. Sloane

Also, the number of 3-connected quadrangulations without separating 4-cycles (up to orientation) with n faces. - Andrey Zabolotskiy, Sep 20 2019

			G.f. = x^6 + x^8 + x^9 + 2*x^10 + 2*x^11 + 9*x^12 + 11*x^13 + 37*x^14 + ...

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).

C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
J. A. D. Cameron, Searching for Squared Squares, USYD Master of Science Thesis (1976). Rare Books & Special Collections Fisher Library, Sydney University.
J. A. D. Cameron, Table 7.2 - listing of tri-connected planar graphs by edge, from the thesis - this was the first count of order 20 (22131). Photo by Stuart E Anderson.
G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs
M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory Series B 66:1 (1996), 87-122.
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
N. D. Kazarinoff and R. Weitzenkamp, Squaring rectangles and squares, Amer. Math. Monthly, 80 (1973), 877-888.

Cf. A002840, A002841, A113201, A078666, A007022, A113205, A338511.

A113204 Same as A007022, but with orientation-reversing isomorphisms forbidden.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 4, 3, 15, 25, 92, 234, 803, 2469, 8512, 28290, 98148, 338673, 1188338, 4180854, 14840031, 52904562, 189724510, 683384218, 2472961423, 8984888982, 32772085447, 119963084542, 440623586740, 1623555117611

Offset: 1

N. J. A. Sloane, Jan 07 2006

nonn

G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54. doi:10.1016/j.disc.2005.10.005
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs

Cf. A113201, A078666, A007022, A002880, A113202, A113205.

a(29) corrected by Brendan McKay, Jun 22 2006
Leading zeros prepended by Max Alekseyev, Sep 13 2016
Offset corrected by Andrey Zabolotskiy, Feb 09 2018

A113205 Same as A002880, but with orientation-reversing isomorphisms forbidden.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 12, 16, 59, 133, 445, 1248, 4162, 13014, 43474, 143304, 484444, 1639388, 5617205, 19332596, 67048051, 233691112, 819121608, 2884443024, 10204104900, 36247138920, 129264732757, 462661038926, 1661637913984

Offset: 6

N. J. A. Sloane, Jan 07 2006

nonn

G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs

Cf. A113201, A078666, A007022, A002880, A113202, A113203, A113204.

A113202 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, with orientation-reversing isomorphisms forbidden.

Original entry on oeis.org

1, 1, 2, 3, 10, 21, 83, 298, 1339, 6049, 29765, 148842, 770267, 4054539, 21743705, 118237471, 651370528, 3628421181, 20416662314, 115919209155, 663548898942, 3826577783917, 22217382001865, 129800215435088

Offset: 4

N. J. A. Sloane, Jan 07 2006

nonn

G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs

Cf. A113201, A078666, A007022, A002880, A113203, A113204, A113205.

A113203 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, minimal degree 3, with orientation-reversing isomorphisms forbidden.

Original entry on oeis.org

1, 0, 1, 1, 4, 3, 16, 26, 99, 256, 895, 2789, 9740, 32799, 115024, 401180, 1421170, 5046539, 18066772, 64940825, 234712099, 851801048, 3104690139, 11358900851, 41710948878, 153684688127, 568079430741, 2106188450292

Offset: 8

N. J. A. Sloane, Jan 07 2006

nonn

G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs

Cf. A113201, A078666, A007022, A002880, A113202, A113204, A113205.

cp's OEIS Frontend

A078666 Number of isomorphism classes of simple quadrangulations of the sphere having n+2 vertices and n faces, minimal degree 3, with orientation-reversing isomorphisms permitted.

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A002880 Number of 3-connected nets with n edges.

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A113204 Same as A007022, but with orientation-reversing isomorphisms forbidden.

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A113205 Same as A002880, but with orientation-reversing isomorphisms forbidden.

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A113202 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, with orientation-reversing isomorphisms forbidden.

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A113203 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, minimal degree 3, with orientation-reversing isomorphisms forbidden.

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Author

Keywords

Links

Crossrefs