A006406 -id:A006406 - cp's OEIS frontend

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

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

A342059 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 up to orientation preserving isomorphisms, n >= 3, k=2..2*n-4.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 2, 1, 1, 3, 17, 31, 22, 6, 2, 1, 4, 42, 157, 318, 265, 123, 26, 6, 1, 6, 87, 576, 2128, 4009, 4055, 2332, 804, 147, 17, 1, 7, 161, 1664, 9659, 31252, 59244, 66289, 46521, 20604, 5743, 892, 73, 1, 9, 286, 4151, 34700, 168757, 505410, 952044, 1156127, 931227, 506318, 183980, 43180, 5876, 389

Offset: 3

Andrew Howroyd, Mar 27 2021

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,  5,   2,    1;
  1, 3, 17,  31,   22,    6,    2;
  1, 4, 42, 157,  318,  265,  123,   26,   6;
  1, 6, 87, 576, 2128, 4009, 4055, 2332, 804, 147, 17;
  ...

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 A342058.

Cf. A006406 (by edges), A239893 (3-connected), A342060.

T(n,2) = 1.

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

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

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

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

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A342059 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 up to orientation preserving isomorphisms, n >= 3, k=2..2*n-4.

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

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

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