Gunnar Brinkmann - cp's OEIS frontend

A299116 The number of sparse union-closed sets. That is, the number of union-closed sets on n elements containing the empty set and the universe, such that in average each set (not counting the empty set) has at most n/2 elements.

Original entry on oeis.org

0, 0, 0, 2, 27, 3133, 5777931

Offset: 1

Gunnar Brinkmann, Feb 05 2018

If there is a counterexample to the union-closed set conjecture, it is a sparse union-closed set.

G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, arXiv:1701.03751 [math.CO], 2017.
G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, Journal of Integer Sequences, Vol.21 (2018), Article 18.1.7.

Cf. A108798, A102894, A193674, A102896.

A111357 Numbers of planar triangulations with minimum degree 5 and without separating 3-cycles - that is 3-cycles where the interior and exterior contain at least one vertex.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 12, 23, 73, 191, 649, 2054, 7209, 24963, 89376, 320133, 1160752, 4218225, 15414908, 56474453, 207586410, 764855802, 2825168619, 10458049611, 38795658003, 144203518881, 537031911877, 2003618333624, 7488436558647

Offset: 12

Gunnar Brinkmann, Nov 07 2005

nonn

			The icosahedron is the smallest triangulation with minimum degree 5 and it doesn't contain any separating triangles. Examples can easily be seen as 2D and 3D pictures using the program CaGe cited above.

G. Brinkmann, CaGe.
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]
G. Brinkmann and Brendan D. McKay, Construction of planar triangulations with minimum degree 5 , Disc. Math. vol 301, iss. 2-3 (2005) 147-163.

Cf. A081621, A007894.

A111358 Numbers of planar triangulations with minimum degree 5 and without separating 3- or 4-cycles - that is 3- or 4-cycles where the interior and exterior contain at least one vertex.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 12, 23, 71, 187, 627, 1970, 6833, 23384, 82625, 292164, 1045329, 3750277, 13532724, 48977625, 177919099, 648145255, 2368046117, 8674199554, 31854078139, 117252592450, 432576302286, 1599320144703, 5925181102878

Offset: 12

Gunnar Brinkmann, Nov 07 2005

nonn

A006791 and this sequence are the same sequence. The correspondence is just that these objects are planar duals of each other. But the offset and step are different: if the cubic graph has 2*n vertices, the dual triangulation has n+2 vertices. - Brendan McKay, May 24 2017

Also the number of 5-connected triangulations on n vertices. - Manfred Scheucher, Mar 17 2023

			The icosahedron is the smallest triangulation with minimum degree 5 and it doesn't contain any separating 3- or 4-cycles. Examples can easily be seen as 2D and 3D pictures using the program CaGe cited above.

G. Brinkmann, CaGe.
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]
G. Brinkmann and Brendan D. McKay, Construction of planar triangulations with minimum degree 5 , Disc. Math. vol 301, iss. 2-3 (2005) 147-163.
D. A. Holton and B. D. McKay, The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices, J. Combinat. Theory B vol 45, iss. 3 (1988) 305-319.
D. A. Holton and B. D. McKay, Erratum, J. Combinat. Theory B vol 47, iss. 2 (1989) 248.

Cf. A081621, A007894, A006791, A000109, A007021, A361578.

A111361 The number of 4-regular plane graphs with n vertices with all faces 3-gons or 4-gons.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 5, 2, 8, 5, 12, 8, 25, 13, 30, 23, 51, 33, 76, 51, 109, 78, 144, 106, 218, 150, 274, 212, 382, 279, 499, 366, 650, 493, 815, 623, 1083, 800, 1305, 1020, 1653, 1261, 2045, 1554, 2505, 1946, 3008, 2322, 3713, 2829, 4354, 3418, 5233, 4063, 6234

Offset: 2

Gunnar Brinkmann, Nov 07 2005

nonn

These are the 4-regular graphs corresponding to the 3-regular fullerenes. Only the two smallest possible face sizes are allowed. The numbers up to a(33) have been checked by 2 independent programs. Further numbers have not been checked independently.

			From _Allan Bickle_, May 13 2024: (Start)
The smallest example (n=6) is the octahedron (only 3-gons).
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)

Andrey Zabolotskiy, Table of n, a(n) for n = 2..131 (from Hasheminezhad & McKay)
G. Brinkmann, O. Heidemeier and T. Harmuth, The construction of cubic and quartic planar maps with prescribed face degrees, Discrete Applied Mathematics 128: 541-554, (2003).
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]
Michel-Marie Deza, Mathieu Dutour Sikiric and Mikhail Ivanovitch Shtogrin, Geometric Structure of Chemistry-Relevant Graphs, Springer, 2015; see Sec. 4.4.
Mathieu Dutour Sikiric and Michel Deza, 4-regular and self-dual analogs of fullerenes, arXiv:0910.5323 [math.GT], 2009.
Mahdieh Hasheminezhad and Brendan D. McKay, Recursive generation of simple planar quadrangulations with vertices of degree 3 and 4, Discussiones Mathematicae Graph Theory, 30 (2010), 123-136.
T. Tarnai, F. Kovács, P. W. Fowler and S. D. Guest, Wrapping the cube and other polyhedra, Proc. Roy. Soc. A 468(2145) (2012), 2652-2666. DOI: 10.1098/rspa.2012.0116.

Cf. A007894.
Cf. A167156, A167157, A167158, A167159.
Cf. A007022, A072552, A078666, A292515 (4-regular planar graphs with restrictions).

Leading zeros prepended, terms a(34) and beyond added from the book by Deza et al. (except for a(60) from the paper by Brinkmann et al.) by Andrey Zabolotskiy, Oct 09 2021

A108071 Number of inner dual graphs of planar polyhexes with n hexagons.

Original entry on oeis.org

1, 1, 2, 4, 8, 21, 53, 151, 458, 1477, 4918, 16956, 59494, 212364, 766753, 2796876, 10284793, 38096072, 141998218, 532301941, 2005638293, 7592441954, 28865031086, 110174528925, 422064799013, 1622379252093

Offset: 1

Gunnar Brinkmann, Jun 05 2005

nonn

			For n = 4, the a(n) = 4 graphs are: the 4-path, which is the inner dual of 4 polyhexes out of A018190(4) = 7 (each of the others is an inner dual of a single polyhex); the paw graph; the diamond graph; the claw graph.

Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A constructive enumeration of fusenes and benzenoids, Journal of Algorithms, Volume 45, Issue 2, November 2002, Pages 155-166.
Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons, J. Chem. Inf. Comput. Sci. 2003, 43, 3, 842-851.

Cf. A018190, A108070, A108072.

Name corrected by Andrey Zabolotskiy, Oct 01 2022

A108072 Number of inner dual graphs of planar polyhexes with n hexagons having no nontrivial symmetry.

Original entry on oeis.org

1, 0, 0, 0, 0, 5, 22, 90, 342, 1247, 4491, 16095, 57906, 209170, 760830, 2784913, 10262649, 38051063, 141914613, 532131882, 2005320952, 7591794561, 28863820538, 110172051829, 422060152511, 1622369728951

Offset: 1

Gunnar Brinkmann, Jun 05 2005

nonn

Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A constructive enumeration of fusenes and benzenoids, Journal of Algorithms, Volume 45, Issue 2, November 2002, Pages 155-166.
Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons, J. Chem. Inf. Comput. Sci. 2003, 43, 3, 842-851.

Cf. A018190, A108070, A108071.

Name corrected by Andrey Zabolotskiy, Oct 01 2022

A108070 Number of fusenes with n hexagons.

Original entry on oeis.org

1, 1, 3, 7, 22, 82, 339, 1505, 7036, 33836, 166246, 829987, 4197273, 21456444, 110716585, 576027737, 3018986040, 15927330105, 84530870455, 451069339063, 2418927725532, 13030938290472, 70492771581350, 382816374644336, 2086362209298079, 11408580755666756

Offset: 1

Gunnar Brinkmann, Jun 05 2005

nonn

Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A constructive enumeration of fusenes and benzenoids, Journal of Algorithms 45 (2002), pp. 155-166.
Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons, J. Chem. Inf. Comput. Sci., vol. 43 (2003) 842-851.
Eric Weisstein's World of Mathematics, Fusene

Cf. A018190, A108071, A108072.

A046880 Number of isolated-pentagon (IPR) fullerenes with 2n vertices (or carbon atoms).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 2, 5, 7, 9, 24, 19, 35, 46, 86, 134, 187, 259, 450, 616, 823, 1233, 1799, 2355, 3342, 4468, 6063, 8148, 10774, 13977, 18769, 23589, 30683, 39393, 49878, 62372, 79362, 98541, 121354, 151201, 186611, 225245, 277930, 335569

Offset: 30

Gunnar Brinkmann and A. Dress (dress(AT)mathematik.uni-bielefeld.de)

Enantiomorphic pairs are regarded as the same here. Cf. A086423.

P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Cambridge Univ. Press, 1995, see p. 33.

Jan Goedgebeur, Table of n, a(n) for n = 30..200, terms a(30)..a(120) from Gunnar Brinkmann.
Gunnar Brinkmann and Andreas W. M. Dress, A constructive enumeration of fullerenes, J. Algorithms 23 (1997), no. 2, 345-358.
Gunnar Brinkmann, Jan Goedgebeur, Brendan D. McKay, The Generation of Fullerenes, arXiv:1207.7010v1 [math.CO], 2012.
Gunnar Brinkmann, Andreas Dress, fullgen.
Gunnar Brinkmann, Jan Goedgebeur, Brendan D. McKay, buckygen.
CombOS - Combinatorial Object Server, generate fullerenes
House of Graphs, Fullerenes.
A. M. Livshits and Yu. E. Lozovik, Cut-and-unfold approach to fullerene enumeration, J. Chem. Inf. Comput. Sci., (2004), vol. 44, 1517-1520.

Cf. A086423, A007894, A057210.

Added a(121)-a(200). a(30)-a(190) is independently confirmed by buckygen and fullgen, while a(191)-a(200) was only computed by buckygen. - Jan Goedgebeur, Aug 08 2012

A004066 Number of simple regular trivalent bicolored graphs with 2n nodes.

Original entry on oeis.org

0, 0, 1, 1, 2, 6, 14, 41, 157, 725, 4196, 29817, 246646, 2297088, 23503564, 260265650, 3090341095, 39101587595, 524783295041, 7443251159470, 111222017297268, 1746166043555813, 28734210790531045, 494526547845483641, 8883866458982018870, 166286444108288113541, 3237719185652343485853, 65477290060076644381373

Offset: 1

Gunnar Brinkmann, Brendan McKay and Eric Rogoyski

nonn

Sean A. Irvine, On the difference between A004066 and A008325
Patric R. J. Östergård, Classifying generalized Howell designs, Designs Codes Cryptog. (2025). See Sect. 3.2.1.

Cf. A000512, A000840, A008325 (bipartite), A006823 (connected).

a(n) = (A000840(n) + A000512(n))/2. - Andrew Howroyd, Apr 01 2020

a(1)-a(2) prepended and terms a(15) and beyond from Andrew Howroyd, Apr 01 2020

A007894 Number of fullerenes with 2n vertices (or carbon atoms).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 6, 15, 17, 40, 45, 89, 116, 199, 271, 437, 580, 924, 1205, 1812, 2385, 3465, 4478, 6332, 8149, 11190, 14246, 19151, 24109, 31924, 39718, 51592, 63761, 81738, 99918, 126409, 153493, 191839, 231017, 285914, 341658, 419013

Offset: 10

Boris Shraiman (boris(AT)physics.att.com), Gunnar Brinkmann and A. Dress (dress(AT)mathematik.uni-bielefeld.de)

Enantiomorphic pairs are regarded as the same here. Cf. A057210.

Contradictory results from the program "buckygen" from Brinkmann et al. (2012) and the program "fullgen" from Brinkmann and Dress (1997) led to the detection of a non-algorithmic error in fullgen. This bug has now been fixed and the results are in complete agreement. a(10)-a(190) were independently confirmed by buckygen and fullgen, while a(191)-a(200) were computed only by buckygen. - Jan Goedgebeur, Aug 08 2012

A. T. Balaban, X. Liu, D. J. Klein, D. Babic, T. G. Schmalz, W. A. Seitz and M. Randic, "Graph invariants for fullerenes", J. Chem. Inf. Comput. Sci., vol. 35 (1995) 396-404.
M. Deza, M. Dutour and P. W. Fowler, Zigzags, railroads and knots in fullerenes, J. Chem. Inf. Comput. Sci., 44 (2004), 1282-1293.
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Cambridge Univ. Press, 1995, see p. 32.
P. W. Fowler, D. E. Manolopoulos and R. P. Ryan, "Isomerization of fullerenes", Carbon, 30 1235 1992.
A. M. Livshits and Yu. E. Lozovik, Cut-and-unfold approach to Fullerene enumeration, J. Chem. Inf. Comput. Sci., 44 (2004), 1517-1520.
Milicevic, A., and N. Trinajstic. "Combinatorial enumeration in chemistry." Chapter 8 in Chemical Modelling: Application and Theory, Vol. 4 (2006): 405-469.
M. Petkovsek and T. Pisanski, Counting disconnected structures: chemical trees, fullerenes, I-graphs and others, Croatica Chem. Acta, 78 (2005), 563-567.

Jan Goedgebeur, Table of n, a(n) for n = 10..200.
Gunnar Brinkmann, Jan Goedgebeur, and Brendan D. McKay, The Generation of Fullerenes, arXiv:1207.7010 [math.CO], 2012.
Gunnar Brinkmann and Andreas Dress, fullgen.
Gunnar Brinkmann and Andreas W. M. Dress, A constructive enumeration of fullerenes, Journal of Algorithms, Vol. 23, No. 2 (1997), 345-358.
Gunnar Brinkmann, Jan Goedgebeur, and Brendan D. McKay, buckygen.
CombOS - Combinatorial Object Server, generate fullerenes
Philip Engel and Peter Smillie, The number of non-negative curvature triangulations of S^2, arXiv:1702.02614 [math.GT], 2017.
Paul Gailiunas, Kagome from Deltahedra, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 337-344.
Jan Goedgebeur and Brendan D. McKay, Fullerenes with distant pentagons, arXiv:1508.02878 [math.CO], (12-August-2015).
House of Graphs, Fullerenes.
Diaaeldin Taha, Wei Zhao, J. Maxwell Riestenberg, and Michael Strube, Normed Spaces for Graph Embedding, arXiv:2312.01502 [cs.LG], 2023. See p. 22.
Eric Weisstein's World of Mathematics, Fullerene
Wikipedia, Fullerene

Cf. A057210, A046880.

a(n) = (809/2612138803200)*sigma_9(n) + O(n^8) where sigma_9(n) is the ninth divisor power sum, cf. A013957. - Philip Engel, Nov 29 2017

Corrected a(68)-a(100) and added a(101)-a(200). - Jan Goedgebeur, Aug 08 2012

cp's OEIS Frontend

User: Gunnar Brinkmann

A299116 The number of sparse union-closed sets. That is, the number of union-closed sets on n elements containing the empty set and the universe, such that in average each set (not counting the empty set) has at most n/2 elements.

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A111357 Numbers of planar triangulations with minimum degree 5 and without separating 3-cycles - that is 3-cycles where the interior and exterior contain at least one vertex.

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A111358 Numbers of planar triangulations with minimum degree 5 and without separating 3- or 4-cycles - that is 3- or 4-cycles where the interior and exterior contain at least one vertex.

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A111361 The number of 4-regular plane graphs with n vertices with all faces 3-gons or 4-gons.

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A108071 Number of inner dual graphs of planar polyhexes with n hexagons.

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A108072 Number of inner dual graphs of planar polyhexes with n hexagons having no nontrivial symmetry.

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A108070 Number of fusenes with n hexagons.

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A046880 Number of isolated-pentagon (IPR) fullerenes with 2n vertices (or carbon atoms).

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A004066 Number of simple regular trivalent bicolored graphs with 2n nodes.

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A007894 Number of fullerenes with 2n vertices (or carbon atoms).

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