A006407 -id:A006407 - cp's OEIS frontend

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.

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).

A006403 Number of unsensed 2-connected planar maps with n edges.

Original entry on oeis.org

0, 1, 2, 3, 6, 15, 36, 114, 396, 1565, 6756, 31563, 154370, 785113, 4099948, 21870704, 118624544, 652485364, 3631820462, 20426666644, 115949791342, 663640383400, 3826858500878, 22218232389849, 129802836253994

Offset: 1

N. J. A. Sloane

nonn

The maps considered here may include 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).
Timothy R. Walsh, personal communication.

Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, p. 41.

Row sums of A379432.

Cf. A000139 (rooted), A006385, A006402 (sensed), A006407 (without parallel edges), A006444 (achiral).

a(n) = (A006402(n) + A006444(n))/2. - Andrew Howroyd, Jan 13 2025

a(23)-a(25) added by Andrew Howroyd, Jan 13 2025

A006406 Number of sensed 2-connected simple planar maps with n edges.

Original entry on oeis.org

1, 1, 2, 4, 9, 24, 81, 274, 1071, 4357, 18416, 80040, 356109, 1610910, 7399114

Offset: 3

N. J. A. Sloane

A simple planar map is a planar map without loops or parallel edges.

Equivalently, a(n) is the number of embeddings on the sphere of 2-connected planar graphs with n edges up to orientation preserving isomorphisms. - Andrew Howroyd, Mar 27 2021

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. A006402, A006404, A006407 (unsensed case), A342059, A379437 (rooted).

a(n) = Sum_{k=3..n} A342059(k, n+2-k). - Andrew Howroyd, Mar 27 2021

a(11) and a(12) from Sean A. Irvine, Apr 02 2017

a(13)-a(17) from Andrew Howroyd, Mar 27 2021

A378076 Number of embeddings on the sphere of 2-connected homeomorphically irreducible planar graphs with n edges.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 16, 37, 100, 293, 888, 2822, 9305, 31274, 106972, 370828, 1298058, 4582413, 16289759, 58259600, 209465186

Offset: 6

Andrew Howroyd, Nov 15 2024

Homeomorphically irreducible means each vertex has a degree of at least 3.

Antidiagonal sums of A378075.

Cf. A002840, A006407, A187928.

a(n) = A002840(n) + A187928(n).

a(n) = Sum_{k=4..n-2} A378075(k, n+2-k).

A379437 Number of rooted 2-connected simple planar maps with n edges.

Original entry on oeis.org

1, 1, 6, 16, 71, 267, 1162

Offset: 3

Andrew Howroyd, Jan 16 2025

A simple planar map is a planar map without loops or parallel edges.

Cf. A000139, A000168, A006406 (sensed), A006407 (unsensed).

cp's OEIS Frontend

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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A006403 Number of unsensed 2-connected planar maps with n edges.

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A006406 Number of sensed 2-connected simple planar maps with n edges.

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A378076 Number of embeddings on the sphere of 2-connected homeomorphically irreducible planar graphs with n edges.

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A379437 Number of rooted 2-connected simple planar maps with n edges.

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