A006395 -id:A006395 - cp's OEIS frontend

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).

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).

A006394 Number of sensed planar maps with n edges and without loops or parallel edges.

Original entry on oeis.org

1, 1, 1, 3, 5, 15, 52, 213, 1002, 5167, 27967, 158447, 926786

Offset: 0

N. J. A. Sloane

The planar maps considered here are connected. A planar map without loops or parallel edges is called simple. - Andrew Howroyd, Jan 16 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.

Cf. A006384, A006390, A006395 (unsensed), A022558 (rooted).

a(9)-a(12) from Sean A. Irvine, Mar 30 2017

a(0)=1 prepended by Andrew Howroyd, Jan 16 2025

A372892 Total number of unlabeled simple maps on the sphere with n vertices.

Original entry on oeis.org

1, 1, 2, 6, 25, 179, 2014, 31178, 580555, 12046072, 267836680, 6258809085, 151983244000, 3807081193879

Offset: 1

Eric W. Weisstein, May 19 2024

Computed by Brendan McKay.

a(n) is the same as the total number of distinct spherical drawings of connected graphs with n vertices.

Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Tables 19-22.

Row sums of A384963, column sums of A384850.
Cf. A372853 (uniquely embeddable connected graphs).
Cf. A372854 (largest numbers of planar embeddings for connected graphs).
Cf. A006395 (with n edges), A384965 (sensed version).

a(14) added by Andrew Howroyd, Jun 13 2025

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).

A006401 Number of unsensed simple planar maps with n edges and without vertices of degree 1.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 5, 10, 28, 86, 285, 1055, 4211

Offset: 0

N. J. A. Sloane

A simple planar map is a planar map without loops or parallel edges. - Andrew Howroyd, Jan 16 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.

Cf. A006385, A006389, A006395, A006400 (sensed), A379436 (rooted).

a(11)-a(12) from Sean A. Irvine, Mar 31 2017

a(0)-a(2) prepended by Andrew Howroyd, Jan 16 2025

cp's OEIS Frontend

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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A006394 Number of sensed planar maps with n edges and without loops or parallel edges.

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A372892 Total number of unlabeled simple maps on the sphere with n vertices.

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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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A006401 Number of unsensed simple planar maps with n edges and without vertices of degree 1.

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