A277741 -id:A277741 - cp's OEIS frontend

A006385 Number of unsensed planar maps with n edges.

1, 2, 4, 14, 52, 248, 1416, 9172, 66366, 518868, 4301350, 37230364, 333058463, 3057319072, 28656583950, 273298352168, 2645186193457, 25931472185976, 257086490694917, 2574370590192556, 26010904915620261

Offset: 0

N. J. A. Sloane

The planar maps considered are connected and may contain loops and parallel edges. - Andrew Howroyd, Jan 13 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, personal communication.

Richard Kapolnai, Gabor Domokos, and Timea Szabo, Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes, Periodica Polytechnica Electrical Engineering, 56(1):11-10, 2012. Also arXiv:1206.1698, 2012. See Table 2.
Valery. A. Liskovets, A reductive technique for enumerating nonisomorphic planar maps, Discr. Math., v.156 (1996), 197-217.
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3
Nicholas C. Wormald, Counting unrooted planar maps, Discrete Math. 36 (1981), no. 2, 205-225.

Antidiagonal sums of A277741.
Column k=0 of A379439.
Cf. A000168 (rooted), A006384 (sensed), A006443 (achiral), A006403 (2-connected), A090376.
Cf. A006387 (genus 1), A214814 (genus 2), A214815 (genus 3), A214816.

a(n) = (A006384(n) + A006443(n))/2. - Andrew Howroyd, Jan 13 2025

a(18)-a(19) added by Andrew Howroyd, Jan 13 2025

a(20) added by Andrew Howroyd, Jan 20 2025

A379430 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 14, 23, 14, 3, 6, 42, 108, 108, 42, 6, 14, 140, 501, 761, 501, 140, 14, 34, 473, 2264, 4744, 4744, 2264, 473, 34, 95, 1670, 10087, 27768, 38495, 27768, 10087, 1670, 95, 280, 5969, 44310, 153668, 279698, 279698, 153668, 44310, 5969, 280

Offset: 1

Andrew Howroyd, Jan 13 2025

The planar maps considered are connected and may contain loops and parallel edges.

The number of edges is n + k - 2.

			Array begins:
=========================================================
n\k |  1    2     3      4      5      6      7     8 ...
----+----------------------------------------------------
  1 |  1    1     1      2      3      6     14    34 ...
  2 |  1    2     5     14     42    140    473  1670 ...
  3 |  1    5    23    108    501   2264  10087 44310 ...
  4 |  2   14   108    761   4744  27768 153668 ...
  5 |  3   42   501   4744  38495 279698 ...
  6 |  6  140  2264  27768 279698 ...
  7 | 14  473 10087 153668 ...
  8 | 34 1670 44310 ...
   ...
As a triangle, rows give the number of edges (first row is 0 edges):
   1;
   1,    1;
   1,    2,     1;
   2,    5,     5,     2;
   3,   14,    23,    14,     3;
   6,   42,   108,   108,    42,     6;
  14,  140,   501,   761,   501,   140,    14;
  34,  473,  2264,  4744,  4744,  2264,   473,   34;
  95, 1670, 10087, 27768, 38495, 27768, 10087, 1670, 95;
  ...

Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, pp. 1-10.

Antidiagonal sums are A006384.
Columns 1..2 are A002995, A380237.
Cf. A269920 (rooted), A277741 (unsensed), A379431 (achiral), A342061 (2-connected), A384964 (simple).

A(n,k) = A(k,n).

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).

A380239 Number of unsensed planar maps with n vertices and 2 faces.

Original entry on oeis.org

1, 2, 5, 13, 35, 104, 315, 1021, 3407, 11814, 41893, 151688, 556432, 2063446, 7709381, 28977788, 109421539, 414759097, 1577080457, 6013019088, 22980514005, 88012484058, 337717418145, 1298113689274, 4997561829650, 19267942661664, 74386901833067, 287540841925770

Offset: 1

Andrew Howroyd, Jan 19 2025

nonn

Also by duality the number of unsensed planar maps with n faces and 2 vertices.

The number of edges is n.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column 2 of A277741.

Cf. A380237 (sensed), A380238 (achiral).

G1(n)={my(g=(1-sqrt(1-4*x^2 + O(x*x^n)))/(2*x^2)); ((1 + x/(1-x-x^2*g)^2)^2/(1 - x^2*g^2) - 1)/2 + 1/(1 - x*g) - 1}
G2(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); sum(k=1, n, my(m=1+k%2); -(log(2 - c(k)) + log(1 - x^k*c(m*k)^(2/m)))*eulerphi(k)/k, O(x*x^n))}
seq(n)={Vec(G1(n)+G2(n))/4}

a(n) = (A380237(n) + A380238(n))/2.

A379431 Array read by antidiagonals: A(n,k) is the number of achiral planar maps with n vertices and k faces, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 12, 17, 12, 3, 6, 28, 58, 58, 28, 6, 10, 68, 179, 247, 179, 68, 10, 20, 157, 538, 942, 942, 538, 157, 20, 35, 372, 1531, 3388, 4345, 3388, 1531, 372, 35, 70, 845, 4288, 11424, 18316, 18316, 11424, 4288, 845, 70

Offset: 1

Andrew Howroyd, Jan 14 2025

The planar maps considered are connected and may contain loops and parallel edges.

The number of edges is n + k - 2.

			==================================================
n\k |  1   2    3     4     5     6     7    8 ...
----+---------------------------------------------
  1 |  1   1    1     2     3     6    10   20 ...
  2 |  1   2    5    12    28    68   157  372 ...
  3 |  1   5   17    58   179   538  1531 4288 ...
  4 |  2  12   58   247   942  3388 11424 ...
  5 |  3  28  179   942  4345 18316 ...
  6 |  6  68  538  3388 18316 ...
  7 | 10 157 1531 11424 ...
  8 | 20 372 4288 ...
  ...
As a triangle, rows give the number of edges (first row is 0 edges):
   1;
   1,   1;
   1,   2,    1;
   2,   5,    5,   2;
   3,  12,   17,   12,    3;
   6,  28,   58,   58,   28,    6;
  10,  68,  179,  247,  179,   68,   10;
  20, 157,  538,  942,  942,  538,  157,  20;
  35, 372, 1531, 3388, 4345, 3388, 1531, 372, 35;
  ...

Antidiagonal sums are A006443.
Column 1 is A210736(n-1).
Cf. A269920 (rooted), A277741 (unsensed), A379430 (sensed).

A(n,k) = A(k,n).

A380616 Triangle read by rows: T(n,k) is the number of unsensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 8, 5, 2, 17, 33, 30, 13, 3, 79, 198, 208, 118, 35, 6, 554, 1571, 1894, 1232, 472, 104, 12, 5283, 16431, 21440, 15545, 6879, 1914, 315, 27, 65346, 213831, 296952, 233027, 115134, 37311, 7881, 1021, 65, 966156, 3288821, 4799336, 4019360, 2163112, 787065, 196267, 32857, 3407, 175

Offset: 0

Andrew Howroyd, Jan 28 2025

By duality, also the number of unsensed combinatorial maps with n edges and k faces.

			Triangle begins:
n\k |     1       2       3       4       5      6     7     8   9
----+--------------------------------------------------------------
  0 |     1;
  1 |     1,      1;
  2 |     2,      2,      1;
  3 |     5,      8,      5,      2;
  4 |    17,     33,     30,     13,      3;
  5 |    79,    198,    208,    118,     35,     6;
  6 |   554,   1571,   1894,   1232,    472,   104,   12;
  7 |  5283,  16431,  21440,  15545,   6879,  1914,  315,   27;
  8 | 65346, 213831, 296952, 233027, 115134, 37311, 7881, 1021, 65;
  ...

Row sums are A214816.
Main diagonal is A006082(n+1).
Columns 1..3 are A054499, A380620, A380621.
Cf. A053979 (rooted), A277741 (planar), A380615 (sensed), A380617 (achiral).

T(n,k) = (A380615(n,k) + A380617(n,k))/2.

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).

A379432 Triangle read by rows: T(n,k) is the number of unsensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 3, 1, 1, 4, 13, 13, 4, 1, 1, 5, 29, 44, 29, 5, 1, 1, 7, 51, 139, 139, 51, 7, 1, 1, 8, 92, 370, 623, 370, 92, 8, 1, 1, 10, 147, 913, 2307, 2307, 913, 147, 10, 1, 1, 12, 240, 2048, 7644, 11673, 7644, 2048, 240, 12, 1, 1, 14, 357, 4295, 22344, 50174, 50174, 22344, 4295, 357, 14, 1

Offset: 2

Andrew Howroyd, Jan 14 2025

The maps considered here may include parallel edges.

The number of faces is n + 2 - k.

			Triangle begins:
   1;
   1,  1;
   1,  1,   1;
   1,  2,   2,   1;
   1,  3,   7,   3,    1;
   1,  4,  13,  13,    4,    1;
   1,  5,  29,  44,   29,    5,   1;
   1,  7,  51, 139,  139,   51,   7,   1;
   1,  8,  92, 370,  623,  370,  92,   8,  1;
   1, 10, 147, 913, 2307, 2307, 913, 147, 10, 1;
   ...

Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, pp. 29-36.

Row sums are A006403.

Cf. A082680 (rooted), A342061 (sensed), A212438 (3-connected), A277741, A342060.

T(n,k) = T(n, n+2-k).

cp's OEIS Frontend

A006385 Number of unsensed planar maps with n edges.

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A379430 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces, n >= 1, k >= 1.

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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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A380239 Number of unsensed planar maps with n vertices and 2 faces.

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A379431 Array read by antidiagonals: A(n,k) is the number of achiral planar maps with n vertices and k faces, n >= 1, k >= 1.

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A380616 Triangle read by rows: T(n,k) is the number of unsensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

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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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A379432 Triangle read by rows: T(n,k) is the number of unsensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.

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