A212438 -id:A212438 - 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

A049337 Triangle read by rows: T(n,k) is the number of 3-connected planar graphs (or polyhedra) with n >= 1 nodes and 0 <= k <= C(n,2) edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 8, 11, 8, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 11, 42, 74, 76, 38, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 74, 296, 633, 768, 558, 219, 50

Offset: 1

Brendan McKay

			Triangle begins
  0;
  0,0;
  0,0,0,0;
  0,0,0,1,0,0,0;
  0,0,0,0,0,1,1,0,0,0;
  0,0,0,0,0,0,1,2,2,...;
  ...
From _Hugo Pfoertner_, Nov 24 2020: (Start)
Transposed table:
.
                              Nodes                        Sums
       4  5  6   7   8    9    10     11     12    13  14 |A002840
  Edges-+--+--+---+---+----+-----+------+------+-----+---+|-------
   6 | 1  .  .   .   .    .     .      .      .     .   . |      1
   7 | .  .  .   .   .    .     .      .      .     .   . |      0
   8 | .  1  .   .   .    .     .      .      .     .   . |      1
   9 | .  1  1   .   .    .     .      .      .     .   . |      2
  10 | .  .  2   .   .    .     .      .      .     .   . |      2
  11 | .  .  2   2   .    .     .      .      .     .   . |      4
  12 | .  .  2   8   2    .     .      .      .     .   . |     12
  13 | .  .  .  11  11    .     .      .      .     .   . |     22
  14 | .  .  .   8  42    8     .      .      .     .   . |     58
  15 | .  .  .   5  74   74     5      .      .     .   . |    158
  16 | .  .  .   .  76  296    76      .      .     .   . |    448
  17 | .  .  .   .  38  633   633     38      .     .   . |   1342
  18 | .  .  .   .  14  768  2635    768     14     .   . |   4199
  19 | .  .  .   .   .  538  6134   6134    558     .   . |  13384
  20 | .  .  .   .   .  219  8822  25626   8822   219   . |  43708
  21 | .  .  .   .   .   50  7916  64439  64439  7916  50 | 144810
  .. | .  .  .   .   .    .    ..     ..     ..    ..  .. |     ..
     ---+--+--+---+---+----+-----+------+-------+----+---+
  Sums 1  2  7  34 257 2606 32300 440564 6384634 .. A000944
(End)

Sean A. Irvine, Table of n, a(n) for n = 1..469; rows 1..14 flattened

Cf. A000944, A002840, A021103, A003094, A049334.
A049337, A058787, A212438 are all versions of the same triangle.
Cf. A058788.

Missing zeros inserted by Sean A. Irvine, Jul 29 2021

A342060 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of 2-connected planar graphs with n nodes and k faces, n >= 3, k=2..2*n-4.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 13, 21, 16, 5, 2, 1, 4, 29, 94, 183, 154, 76, 18, 5, 1, 6, 59, 328, 1146, 2114, 2144, 1246, 447, 88, 14, 1, 7, 104, 915, 5046, 16009, 30183, 33719, 23749, 10585, 3017, 489, 50, 1, 9, 181, 2239, 17876, 85550, 254831, 478913, 581324, 468388, 255156, 93028, 22077, 3071, 233

Offset: 3

Andrew Howroyd, Mar 27 2021

Equivalently, T(n,k) is the number of unsensed 2-connected planar maps with n vertices and k faces.
The number of edges is n+k-2.
Terms of this sequence can be computed using the tool "plantri". The expanded reference gives rows 3..15 of this table.

			Triangle begins:
  1;
  1, 1,  1;
  1, 2,  4,   2,    1;
  1, 3, 13,  21,   16,    5,    2;
  1, 4, 29,  94,  183,  154,   76,   18,   5;
  1, 6, 59, 328, 1146, 2114, 2144, 1246, 447, 88, 14;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..171 (rows 3..15)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Tables 23-26.

Row sums are A034889.

Cf. A006407 (by edges), A212438 (3-connected), A342059.

T(n,2) = 1.

T(n,3) = A253186(n-2).

A239893 Irregular triangle read by rows: T(n,k) is the number of sensed 3-connected planar maps with n >= 4 faces and k >= 4 vertices.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 3, 2, 2, 0, 0, 2, 11, 16, 10, 6, 0, 0, 2, 16, 69, 127, 128, 60, 17, 0, 0, 0, 10, 127, 541, 1188, 1441, 1032, 386, 73, 0, 0, 0, 6, 128, 1188, 5096, 11982, 17265, 15466, 8582, 2652, 389, 0, 0, 0, 0, 60, 1441, 11982, 50586, 127765, 206880, 222472, 158057, 71980, 18914, 2274

Offset: 4

N. J. A. Sloane, Apr 03 2014

T(n,k) is the number of polyhedra with n faces and k vertices up to orientation preserving isomorphisms. The number of edges is n+k-2. - Andrew Howroyd, Mar 27 2021

			Triangle begins:
1
0 1 1
0 1 3  2   2
0 0 2 11  16   10     6
0 0 2 16  69  127   128    60     17
0 0 0 10 127  541  1188  1441   1032    386     73
0 0 0  6 128 1188  5096 11982  17265  15466   8582   2652   389
0 0 0  0  60 1441 11982 50586 127765 206880 222472 158057 71980 18914 2274
...

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.
Timothy R. Walsh, Efficient enumeration of sensed planar maps, Discrete Math. 293 (2005), no. 1-3, 263--289. MR2136069 (2006b:05062).
Timothy R. S. Walsh, Counting nonisomorphic three-connected planar maps, J. Combin. Theory Ser. B 32 (1982), no. 1, 33-44.

Row and column sums are A119501.
Main diagonal is A342057.
The unsensed version is A212438.
Cf. A005645 (by edges).

T(n,k) = T(k,n). - Andrew Howroyd, Mar 27 2021

Terms a(67) and beyond from Andrew Howroyd, Mar 27 2021

A384963 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces, n >= 1, k=1..max(1,2*n-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 7, 5, 2, 1, 6, 22, 42, 49, 35, 18, 5, 2, 12, 76, 237, 442, 510, 412, 218, 84, 18, 5, 27, 271, 1293, 3539, 6205, 7482, 6318, 3833, 1623, 485, 88, 14, 65, 1001, 6757, 25842, 63254, 106985, 129782, 115988, 76582, 37421, 13111, 3228, 489, 50

Offset: 1

Andrew Howroyd, Jun 13 2025

Equivalently, T(n,k) is the number of unsensed simple planar maps with n vertices and k faces.
The number of edges is n+k-2.
Terms of this sequence can be computed using the tool "plantri". The expanded reference gives rows 1..14 of this table.

			Triangle begins:
   1;
   1;
   1,   1;
   2,   2,    1,    1,
   3,   7,    7,    5,    2,    1;
   6,  22,   42,   49,   35,   18,    5,    2;
  12,  76,  237,  442,  510,  412,  218,   84,   18,   5;
  27, 271, 1293, 3539, 6205, 7482, 6318, 3833, 1623, 485, 88, 14;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..158 (rows 1..14)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Tables 19-22.

Row sums are A372892.
Antidiagonal sums are A006395.
Columns 1..2 are A006082, A384967.
Cf. A277741 (not necessarily simple), A342060 (2-connected), A212438 (3-connected), A384850 (version by number of edges then vertices), A384964 (sensed version).

A002856 Number of polyhedra with n nodes and n faces.

Original entry on oeis.org

1, 1, 2, 8, 42, 296, 2635, 25626, 268394, 2937495, 33310550, 388431688, 4637550072, 56493493990, 700335433295

Offset: 4

N. J. A. Sloane

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).

P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
G. P. Michon, Counting Polyhedra

One of the main diagonals of A212438.

More terms from Gerard P. Michon, Mar 29 2002

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.

A378075 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of 2-connected homeomorphically irreducible planar graphs with n nodes and k faces, k=4..2n-4.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 3, 3, 2, 0, 0, 3, 11, 18, 10, 5, 0, 0, 3, 19, 77, 134, 123, 50, 14, 0, 0, 0, 13, 146, 603, 1280, 1420, 883, 278, 50, 0, 0, 0, 8, 162, 1409, 6030, 13781, 18404, 14570, 6884, 1772, 233, 0, 0, 0, 0, 83, 1809, 15225, 64502, 158717, 240841, 233286, 144005, 55444, 12077, 1249

Offset: 4

Andrew Howroyd, Nov 15 2024

The number of edges is n + k - 2.

			Triangle begins:
  n\k| 4  5  6   7    8     9    10     11     12     13    14    15   16
-----+--------------------------------------------------------------------
   4 | 1;
   5 | 0, 1, 1;
   6 | 0, 1, 3,  3,   2;
   7 | 0, 0, 3, 11,  18,   10,    5;
   8 | 0, 0, 3, 19,  77,  134,  123,    50,    14;
   9 | 0, 0, 0, 13, 146,  603, 1280,  1420,   883,   278,   50;
  10 | 0, 0, 0,  8, 162, 1409, 6030, 13781, 18404, 14570, 6884, 1772, 233;
  ...

Andrew Howroyd, Table of n, a(n) for n = 4..124 (rows 4..14)

Row sums are A378074.
Antidiagonal sums give A378076.
Cf. A212438, A342060, A378077.

T(n,k) = A212438(n,k) + A378077(n,k).

A384850 Triangle read by rows: T(n,k) is the number of unsensed simple planar maps with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 3, 0, 0, 0, 1, 7, 6, 0, 0, 0, 1, 7, 22, 12, 0, 0, 0, 0, 5, 42, 76, 27, 0, 0, 0, 0, 2, 49, 237, 271, 65, 0, 0, 0, 0, 1, 35, 442, 1293, 1001, 175, 0, 0, 0, 0, 0, 18, 510, 3539, 6757, 3765, 490

Offset: 0

Andrew Howroyd, Jun 13 2025

The planar maps considered here are connected.

The initial terms of this sequence can be computed using the tool "plantri", in particular the command "./plantri -u -v -c1 -p [n]" will compute values for a column.

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 2;
  0, 0, 0, 2, 3;
  0, 0, 0, 1, 7,  6;
  0, 0, 0, 1, 7, 22,  12;
  0, 0, 0, 0, 5, 42,  76,   27;
  0, 0, 0, 0, 2, 49, 237,  271,   65;
  0, 0, 0, 0, 1, 35, 442, 1293, 1001, 175;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..135 (rows 0..15)
Gunnar Brinkmann and Brendan McKay, Plantri (program for generation of certain types of planar graph).

Row sums are A006395.
Column sums are A372892.
Main diagonal is A006082.
Subdiagonal is A384967.
Cf. A054923 (graphs), A277741 (not necessarily simple), A342060 (2-connected), A212438 (3-connected), A384963 (version by number of vertices then faces).

A378077 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of planar graphs with n vertices and k faces having connectivity exactly 2 and minimum vertex degree at least 3, k=6..2n-5.

Original entry on oeis.org

1, 1, 1, 3, 7, 2, 1, 8, 35, 60, 47, 12, 0, 5, 72, 307, 647, 652, 325, 59, 0, 3, 86, 776, 3395, 7647, 9582, 6654, 2442, 368, 0, 0, 45, 1041, 9091, 38876, 94278, 136628, 121204, 64232, 18916, 2363, 0, 0, 18, 827, 14407, 111076, 468211, 1192511, 1937266, 2049784, 1409199, 607746, 150161, 16253

Offset: 6

Andrew Howroyd, Nov 15 2024

The graphs are 2-connected, but not 3-connected. Graphs with minimum degree at least 3 are also called homeomorphically irreducible.

The number of edges is n + k - 2.

			Triangle begins:
  n\k| 6  7   8     9    10     11     12      13      14     15     16    17
-----+------------------------------------------------------------------------
   6 | 1, 1;
   7 | 1, 3,  7,    2;
   8 | 1, 8, 35,   60,   47,    12;
   9 | 0, 5, 72,  307,  647,   652,   325,     59;
  10 | 0, 3, 86,  776, 3395,  7647,  9582,   6654,   2442,   368;
  11 | 0, 0, 45, 1041, 9091, 38876, 94278, 136628, 121204, 64232, 18916, 2363;
  ...

Andrew Howroyd, Table of n, a(n) for n = 6..95 (rows 6..14)

Rows sums are A187927.
Antidiagonals sums give A187928.
Cf. A378075.

T(n,k) = A212438(n,k) - A378075(n,k).

cp's OEIS Frontend

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

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A049337 Triangle read by rows: T(n,k) is the number of 3-connected planar graphs (or polyhedra) with n >= 1 nodes and 0 <= k <= C(n,2) edges.

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A342060 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of 2-connected planar graphs with n nodes and k faces, n >= 3, k=2..2*n-4.

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A239893 Irregular triangle read by rows: T(n,k) is the number of sensed 3-connected planar maps with n >= 4 faces and k >= 4 vertices.

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A384963 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces, n >= 1, k=1..max(1,2*n-4).

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A002856 Number of polyhedra with n nodes and n faces.

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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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A378075 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of 2-connected homeomorphically irreducible planar graphs with n nodes and k faces, k=4..2n-4.

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A384850 Triangle read by rows: T(n,k) is the number of unsensed simple planar maps with n edges and k vertices, 1 <= k <= n+1.

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A378077 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of planar graphs with n vertices and k faces having connectivity exactly 2 and minimum vertex degree at least 3, k=6..2n-5.

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