A049336 -id:A049336 - cp's OEIS frontend

A000944 Number of polyhedra (or 3-connected simple planar graphs) with n nodes.

0, 0, 0, 1, 2, 7, 34, 257, 2606, 32300, 440564, 6384634, 96262938, 1496225352, 23833988129, 387591510244, 6415851530241, 107854282197058

Offset: 1

N. J. A. Sloane

H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, B15.
M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
B. Grünbaum, Convex Polytopes. Wiley, NY, 1967, p. 424.
Y. Y. Prokhorov, ed., Mnogogrannik [Polyhedron], Mathematical Encyclopedia Dictionary, Soviet Encyclopedia, 1988.
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).
G. M. Ziegler, Questions about polytopes, pp. 1195-1211 of Mathematics Unlimited - 2001 and Beyond, ed. B. Engquist and W. Schmid, Springer-Verlag, 2001.

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
A. J. W. Duijvestijn and P. J. Federico, The number of polyhedral (3-connected planar) graphs, Math. Comp. 37 (1981), no. 156, 523-532. MR0243424 (39 #4746).
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
Lukas Finschi, A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids, A dissertation submitted to the Swiss Federal Institute of Technology, Zurich for the degree of Doctor of Mathematics, 2001. See p. 155.
Moritz Firsching, Realizability and inscribability for simplicial polytopes via nonlinear optimization. Math. Program. 166, No. 1-2 (A), 273-295 (2017). Table 1
Komei Fukuda, Hiroyuki Miyata, and Sonoko Moriyama, Complete Enumeration of Small Realizable Oriented Matroids, arXiv:1204.0645 [math.CO], 2012; Discrete Comput. Geom. 49 (2013), no. 2, 359--381. MR3017917. - From _N. J. A. Sloane_, Feb 16 2013
Jorik Jooken, Computer-assisted graph theory: a survey, arXiv:2508.20825 [math.CO], 2025. See Ref. 196 at p. 5.
A. B. Korchagin, Ordering Cellular Spaces with Application to Curves and Knots, Discrete Comput. Geom., 40 (2008), 289-311.
G. P. Michon, Counting Polyhedra
Eric Weisstein's World of Mathematics, Polyhedral Graph

Cf. A005470, A049337, A049334, A003094, A049336, A021103, A005841.

Row sums of A212438.

More terms from Brendan McKay

a(18) from Brendan McKay, Jun 02 2006

A021103 Number of two-connected (or biconnected) planar graphs with n nodes.

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 44, 294, 2893, 36496, 545808, 9029737, 159563559, 2952794985, 56589742050

Offset: 0

Brendan McKay

For n < 3, conventions vary: Read & Wilson set a(2) = 0, but Gagarin et al. set a(2) = 1. - Andrey Zabolotskiy, Jun 07 2023

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998. See p. 229.

A. Gagarin, G. Labelle, P. Leroux, and T. Walsh, Structure and enumeration of two-connected graphs with prescribed three-connected components, Adv. in Appl. Math. 43 (2009), no. 1, pp. 46-74. See (116) on p. 69.

Row sums of A049336.
The labeled version is A096331.
Cf. A000944 (3-connected), A002218, A003094, A005470.

a(12)-a(14) from Gilbert Labelle (labelle.gilbert(AT)uqam.ca), Jan 20 2009
Offset 0 from Michel Marcus, Jun 05 2023
a(2) changed back to 0 by Georg Grasegger and Andrey Zabolotskiy, Jun 07 2023

A049334 Triangle read by rows: T(n, k) is the number of unlabeled connected planar simple graphs with n >= 1 nodes and 0<=k<=3*n-6 edges.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 1, 1, 0, 0, 0, 0, 3, 5, 5, 4, 2, 1, 0, 0, 0, 0, 0, 6, 13, 19, 22, 19, 13, 5, 2, 0, 0, 0, 0, 0, 0, 11, 33, 67, 107, 130, 130, 96, 51, 16, 5, 0, 0, 0, 0, 0, 0, 0, 23, 89, 236, 486, 804, 1112, 1211, 1026, 626, 275, 72, 14, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Brendan McKay

Planar graphs with n >= 3 nodes have at most 3*n-6 edges.

			n\k 0  1  2  3  4  5  6  7  8  9 10 11 12
--:-- -- -- -- -- -- -- -- -- -- -- -- --
1:  1
2:  0  1
3:  0  0  1  1
4:  0  0  0  2  2  1  1
5:  0  0  0  0  3  5  5  4  2  1
6:  0  0  0  0  0  6 13 19 22 19 13  5  2

Georg Grasegger, Table of n, a(n) for n = 1..235 (rows 1..13) (terms n = 1..147 (rows 1..11) from Andrew Howroyd)
F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Amer. Math. Soc. 78 (1955), 445-463. (MR0068198) See page 457, equation (2.9).

Row sums are A003094.
Column sums are A046091.
Cf. A000055, A039735, A049336, A049337, A054924, A288265, A343873 (transpose).

geng -c $n $k:$k | planarg -q | countg -q # Georg Grasegger, Jul 11 2023

T(n, n-1) = A000055(n) and Sum_{k} T(n, k) = A003094(n) if n>=1. - Michael Somos, Aug 23 2015

log(1 + B(x, y)) = Sum{n>0} A(x^n, y^n) / n where A(x, y) = Sum_{n>0, k>=0} T(n,k) * x^n * y^k and similarly B(x, y) with A039735. - Michael Somos, Aug 23 2015

A343869 Number of unlabeled nonseparable (or 2-connected) planar graphs with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 7, 16, 41, 108, 320, 1042, 3575, 13064, 49938, 197729, 805991, 3363084, 14302891, 61813285, 270805177, 1200460492, 5376709415, 24302430375, 110745093999, 508380790741

Offset: 1

Andrew Howroyd, May 04 2021

Terms may be computed using the tools geng and planarg in nauty.

Brendan McKay and Adolfo Piperno, nauty and Traces

Row sums of A343870.
Column sums of A049336(n > 1).
Cf. A002840 (3-connected), A010355, A021103, A046091, A289471, A291841.

# count graphs for the sequence by number of vertices v, sum over v afterwards
geng -C $v $n:$n | planarg -q | countg -q # Georg Grasegger, Jun 05 2023

a(21)-a(26) added by Georg Grasegger, Jun 05 2023

A343870 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) planar graphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 3, 3, 1, 0, 0, 0, 0, 2, 9, 4, 1, 0, 0, 0, 0, 1, 13, 20, 6, 1, 0, 0, 0, 0, 0, 11, 49, 40, 7, 1, 0, 0, 0, 0, 0, 5, 77, 158, 70, 9, 1, 0, 0, 0, 0, 0, 2, 75, 406, 426, 121, 11, 1, 0, 0, 0, 0, 0, 0, 47, 662, 1645, 1018, 189, 13, 1, 0

Offset: 1

Andrew Howroyd, May 04 2021

			Triangle T(n,k) begins (n edges >= 1, k vertices >= 2):
  1;
  0, 0;
  0, 1, 0;
  0, 0, 1, 0;
  0, 0, 1, 1,  0;
  0, 0, 1, 2,  1,  0;
  0, 0, 0, 3,  3,  1,   0;
  0, 0, 0, 2,  9,  4,   1,   0;
  0, 0, 0, 1, 13, 20,   6,   1,   0;
  0, 0, 0, 0, 11, 49,  40,   7,   1,  0;
  0, 0, 0, 0,  5, 77, 158,  70,   9,  1, 0;
  0, 0, 0, 0,  2, 75, 406, 426, 121, 11, 1, 0;
  ...

Georg Grasegger, Table of n, a(n) for n = 1..351 (26 rows) (first 210 terms (20 rows) from Andrew Howroyd)

Row sums are A343869.
Column sums are A021103.
Cf. A049334, A049336 (transpose), A049337, A253186, A339070.

geng -C $k $n:$n | planarg -q | countg -q # Georg Grasegger, Jun 05 2023

T(n, n) = 1 for n >= 3.

T(n, n-1) = A253186(n-3) for n >= 3.

cp's OEIS Frontend

A000944 Number of polyhedra (or 3-connected simple planar graphs) with n nodes.

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A021103 Number of two-connected (or biconnected) planar graphs with n nodes.

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A049334 Triangle read by rows: T(n, k) is the number of unlabeled connected planar simple graphs with n >= 1 nodes and 0<=k<=3*n-6 edges.

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nauty

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A343869 Number of unlabeled nonseparable (or 2-connected) planar graphs with n edges.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

nauty

Extensions

A343870 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) planar graphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

nauty

Formula