A049337 -id:A049337 - 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

A002840 Number of polyhedral graphs with n edges.

Original entry on oeis.org

1, 0, 1, 2, 2, 4, 12, 22, 58, 158, 448, 1342, 4199, 13384, 43708, 144810, 485704, 1645576, 5623571, 19358410, 67078828, 233800162, 819267086, 2884908430, 10204782956, 36249143676, 129267865144, 462669746182, 1661652306539, 5986979643542

Offset: 6

N. J. A. Sloane

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
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).
T. R. S. Walsh, personal communication.

C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
A. J. W. Duijvestijn and P. J. Federico, The number of polyhedral (3-connected planar) graphs, Math. Comp. 37 (1981), no. 156, 523-532.
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
G. P. Michon, Counting Polyhedra - Numericana
Hugo Pfoertner, Unlabeled 3-connected planar graphs for n<=20 edges, list in PARI-readable format.
Eric Weisstein's World of Mathematics, Polyhedral Graph
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices

Column sums of A049337.

Cf. A002841, A000944, A046091, A338511, A343869, A343871.

\\ It is assumed that the 3cp.gp file (from the linked zip archive) has been read before, i.e., \r [path]3cp.gp
for(k=6,#ThreeConnectedData,print1(#ThreeConnectedData[k],", "));
\\ printing of the edge lists of the graphs for n <= 11
print(ThreeConnectedData[6..11]) \\ Hugo Pfoertner, Feb 14 2021

a(30)-a(35) from the Numericana link added by Andrey Zabolotskiy, Jun 13 2020

A212438 Irregular triangle read by rows: T(n,k) is the number of polyhedra with n faces and k vertices (n >= 4, k=4..2n-4).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 2, 0, 0, 2, 8, 11, 8, 5, 0, 0, 2, 11, 42, 74, 76, 38, 14, 0, 0, 0, 8, 74, 296, 633, 768, 558, 219, 50, 0, 0, 0, 5, 76, 633, 2635, 6134, 8822, 7916, 4442, 1404, 233, 0, 0, 0, 0, 38, 768, 6134, 25626, 64439, 104213, 112082, 79773, 36528, 9714, 1249

Offset: 4

N. J. A. Sloane, May 16 2012

Because of duality, T(n,k) = T(k,n). - Ivan Neretin, May 25 2016

The number of edges is n+k-2. - Andrew Howroyd, Mar 27 2021

			Triangle begins:
1
0 1 1
0 1 2  2  2
0 0 2  8 11   8    5
0 0 2 11 42  74   76   38   14
0 0 0  8 74 296  633  768  558  219   50
0 0 0  5 76 633 2635 6134 8822 7916 4442 1404 233
...

Andrew Howroyd, Table of n, a(n) for n = 4..199 (rows 4..17)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Table 9-11.
G. P. Michon, Counting Polyhedra

A049337, A058787, A212438 are all versions of the same triangle.
Row sums (the same as column sums) are A000944.
Main diagonal is A002856.
Cf. A002840 (by edges), A239893.

Terms a(53) and beyond from Andrew Howroyd, Mar 27 2021

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

A000943 Number of combinatorial types of simplicial n-dimensional polytopes with n+3 nodes.

Original entry on oeis.org

1, 2, 5, 8, 18, 29, 57, 96, 183, 318, 604, 1080, 2047, 3762, 7145, 13354, 25471, 48164, 92193, 175780, 337581, 647313, 1246849, 2400828, 4636375, 8956045, 17334785, 33570800, 65108045, 126355319, 245492226, 477284164, 928772631, 1808538336

Offset: 1

N. J. A. Sloane

B. Grünbaum, Convex Polytopes. Wiley, NY, 1967, p. 424.
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).

Cf. A049337, A000944.

with(numtheory); n := 50; for d from 2 to n do C(d) := 0; for h from 1 to d+3 do if (h mod 2 = 1) and (d+3 mod h = 0) then C(d) := C(d) + phi(h) * 2^((d+3)/h); fi; od; C(d) := 2^(floor(d/2)) - floor ((d+4)/2) + C(d)/(4*(d+3)); od: A000943 := n-> eval(C(n));

a[ n_ ] := 2^Floor[ n/2 ]-Floor[ (n+4)/2 ]+(1/(4*(n+3)))*Plus@@Map[ EulerPhi[ # ]*2^((n+3)/#)&, Select[ Divisors[ n+3 ], OddQ ] ]

n=12 term corrected (typo in reference), formula (due to Perles) and more terms from Lukas Finschi (finschi(AT)ifor.math.ethz.ch), Mar 06 2001

A058787 Triangle T(n,k) = number of polyhedra (triconnected planar graphs) with n faces and k vertices, where (n/2+2) <= k <= (2n+8).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 8, 11, 8, 5, 2, 11, 42, 74, 76, 38, 14, 8, 74, 296, 633, 768, 558, 219, 50, 5, 76, 633, 2635, 6134, 8822, 7916, 4442, 1404, 233, 38, 768, 6134, 25626, 64439, 104213, 112082, 79773, 36528, 9714, 1249, 14, 558, 8822, 64439, 268394, 709302

Offset: 4

Gerard P. Michon, Nov 29 2000

Rows are of lengths 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, ... floor(3n/2)-5. See A001651 (this is the sequence of integers not divisible by 3).

			There are 38 polyhedra with 9 faces and 11 vertices, or with 11 faces and 9 vertices.

G. P. Michon, Counting Polyhedra

Cf. A000109, A002856, A000944, A002840, A058786, A058788, A001651.

A049337, A058787, A212438 are all versions of the same triangle.

A058786 Number of n-hedra with 2n-5 vertices or 3n-7 edges (the vertices of these are all of degree 3, except one which is of degree 4). Alternatively, the number of polyhedra with n vertices whose faces are all triangular, except one which is tetragonal.

Original entry on oeis.org

1, 2, 8, 38, 219, 1404, 9714, 70454, 527235, 4037671, 31477887, 249026400, 1994599707, 16147744792, 131959532817, 1087376999834, 9027039627035, 75441790558926, 634311771606750, 5362639252793358, 45565021714371644, 388937603694422120, 3333984869758146814

Offset: 5

Gerard P. Michon, Nov 29 2000

			a(5)=1 because the square pyramid is the only pentahedron with 5=2*5-5 vertices (or 8=3*5-7 edges). Alternatively, a(5)=1 because the square pyramid is the only polyhedron with 5 vertices whose faces are all triangles with only one tetragonal exception.

Andrew Howroyd, Table of n, a(n) for n = 5..500
CombOS - Combinatorial Object Server, generate planar graphs
G. P. Michon, Counting Polyhedra

Column k=4 of A342053.

Cf. A000109, A002856, A000944, A002840, A058787, A058788, A049337.

A342053ColSeq(25,4) \\ See links in A342053. - Andrew Howroyd, Feb 28 2021

Terms a(19) and beyond from Andrew Howroyd, Feb 27 2021

A058788 Triangle T(n,k) = number of polyhedra (triconnected planar graphs) with n edges and k vertices (or k faces), where (n/3+2) <= k <= (2n/3). Note that there is no such k when n=7.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 8, 2, 11, 11, 8, 42, 8, 5, 74, 74, 5, 76, 296, 76, 38, 633, 633, 38, 14, 768, 2635, 768, 14, 558, 6134, 6134, 558, 219, 8822, 25626, 8822, 219, 50, 7916, 64439, 64439, 7916, 50, 4442, 104213, 268394, 104213, 4442, 1404, 112082, 709302, 709302, 112082, 1404, 233, 79773, 1263032, 2937495, 1263032, 79773, 233, 36528, 1556952, 8085725, 8085725, 1556952, 36528, 9714, 1338853, 15535572, 33310550

Offset: 6

Gerard P. Michon, Nov 29 2000

Rows are of lengths 1,0,1,2,1,2,3,2,3,4,3,4,5,4,5,6,5, ... n-1-2*floor((n+2)/3). See A008611. Note the zero length, which means that there are no polyhedra with n=7 edges.

			There are 768 different polyhedra with 18 edges and 9 or 11 faces.

G. P. Michon, Counting Polyhedra

Cf. A000109, A002856, A000944, A002840, A058786, A058787, A049337, A008611.

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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A002840 Number of polyhedral graphs with n edges.

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A212438 Irregular triangle read by rows: T(n,k) is the number of polyhedra with n faces and k vertices (n >= 4, k=4..2n-4).

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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

Formula

A000943 Number of combinatorial types of simplicial n-dimensional polytopes with n+3 nodes.

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Maple

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A058787 Triangle T(n,k) = number of polyhedra (triconnected planar graphs) with n faces and k vertices, where (n/2+2) <= k <= (2n+8).

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A058786 Number of n-hedra with 2n-5 vertices or 3n-7 edges (the vertices of these are all of degree 3, except one which is of degree 4). Alternatively, the number of polyhedra with n vertices whose faces are all triangular, except one which is tetragonal.

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Author

Keywords

Examples

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Programs

PARI

Extensions

A058788 Triangle T(n,k) = number of polyhedra (triconnected planar graphs) with n edges and k vertices (or k faces), where (n/3+2) <= k <= (2n/3). Note that there is no such k when n=7.

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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).

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