A006403 -id:A006403 - 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

A006402 Number of sensed 2-connected (nonseparable) planar maps with n edges.

Original entry on oeis.org

1, 2, 3, 6, 16, 42, 151, 596, 2605, 12098, 59166, 297684, 1538590, 8109078, 43476751, 236474942, 1302680941, 7256842362, 40832979283, 231838418310, 1327095781740, 7653155567834, 44434752082990, 259600430870176, 1525366978752096, 9010312253993072, 53485145730576790

Offset: 2

N. J. A. Sloane

nonn

Some people begin this 2,1,2,3,6,..., others begin it 0,1,2,3,6,....

The dual of a nonseparable map is nonseparable, so the class of all nonseparable planar maps is self-dual. The maps considered here are unrooted and sensed and may include loops and parallel edges. - Andrew Howroyd, Mar 29 2021

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

Andrew Howroyd, Table of n, a(n) for n = 2..500
V. A. Liskovets, T. R. S. Walsh, The enumeration of nonisomorphic 2-connected planar maps, Canad. J. Math. 35 (1983), no. 3, 417-435.
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices

Row sums of A342061.

Cf. A000087 (with distinguished faces), A000139 (rooted), A005645, A006403 (unsensed), A006406 (without loops or parallel edges).

\\ here r(n) is A000139(n-1).
r(n)={4*binomial(3*n,n)/(3*(3*n-2)*(3*n-1))}
a(n)={(r(n) + sumdiv(n, d, if(dAndrew Howroyd, Mar 29 2021

Terms a(23) and beyond from Andrew Howroyd, Mar 29 2021

A006407 Number of unsensed 2-connected simple planar maps with n edges.

Original entry on oeis.org

1, 1, 2, 4, 8, 20, 58, 177, 630, 2410, 9772, 41423, 181586, 814412, 3722445

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. - 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. A006403, A006405, A006406 (sensed case), A342060, A379437 (rooted).

a(n) = Sum_{k=3..n} A342060(k, n+2-k). - Andrew Howroyd, Mar 27 2021

a(11) and a(12) from Sean A. Irvine, Apr 03 2017

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

A006405 Number of unsensed 2-connected maps with n edges and without faces of degree 2.

Original entry on oeis.org

1, 1, 2, 5, 9, 24, 70, 222, 785, 3055

Offset: 3

N. J. A. Sloane

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, A006403, A006405 (sensed).

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

A006444 Number of achiral 2-connected planar maps with n edges.

Original entry on oeis.org

0, 1, 2, 3, 6, 14, 30, 77, 196, 525, 1414, 3960, 11056, 31636, 90818, 264657, 774146, 2289787, 6798562, 20354005, 61164374, 184985060, 561433922, 1712696708, 5241637812

Offset: 1

N. J. A. Sloane

nonn

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

Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, p. 44.

Cf. A006402 (sensed), A006403 (unsensed), A006443 (connected), A006445 (3-connected).

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).

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A006385 Number of unsensed planar maps with n edges.

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A006402 Number of sensed 2-connected (nonseparable) planar maps with n edges.

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A006407 Number of unsensed 2-connected simple planar maps with n edges.

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A006405 Number of unsensed 2-connected maps with n edges and without faces of degree 2.

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A006444 Number of achiral 2-connected planar maps with n edges.

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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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