A379432 -id:A379432 - cp's OEIS frontend

A277741 Array read by antidiagonals: A(n,k) is the number of unsensed planar maps with n vertices and k faces, n >= 1, k >= 1.

1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 13, 20, 13, 3, 6, 35, 83, 83, 35, 6, 12, 104, 340, 504, 340, 104, 12, 27, 315, 1401, 2843, 2843, 1401, 315, 27, 65, 1021, 5809, 15578, 21420, 15578, 5809, 1021, 65, 175, 3407, 24299, 82546, 149007, 149007, 82546, 24299, 3407, 175

Offset: 1

N. J. A. Sloane, Nov 07 2016

A(n,k) is also the number of multiquadrangulations of the sphere with n stable equilibria and k unstable equilibria.
From Andrew Howroyd, Jan 13 2025: (Start)
The planar maps considered are connected and may contain loops and parallel edges.
The number of edges is n + k - 2. (End)

			The array begins:
   1,    1,    1,     2,     3,     6,   12,   27, 65, ...
   1,    2,    5,    13,    35,   104,  315, 1021, ...
   1,    5,   20,    83,   340,  1401, 5809, ...
   2,   13,   83,   504,  2843, 15578, ...
   3,   35,  340,  2843, 21420, ...
   6,  104, 1401, 15578, ...
  12,  315, 5809, ...
  27, 1021, ...
  65, ...
  ...
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,   13,   20,    13,     3;
   6,   35,   83,    83,    35,    6;
  12,  104,  340,   504,   340,   104,   12;
  27,  315, 1401,  2843,  2843,  1401,  315,   27;
  65, 1021, 5809, 15578, 21420, 15578, 5809, 1021, 65;
  ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, chapter 5.

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 [cs.DM], 2012. See Table 1.
Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, pp. 11-20.
Nicholas C. Wormald, Counting unrooted planar maps, Discrete Math. 36 (1981), no. 2, 205-225.

Antidiagonal sums are A006385.
Rows 1..2 (equally, columns 1..2) are A006082, A380239.
Cf. A269920 (rooted), A379430 (sensed), A379431 (achiral), A379432 (2-connected), A384963 (simple).

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

A(n,k) = (A379430(n,k) + A379431(n,k))/2. - Andrew Howroyd, Jan 14 2025

Missing terms inserted and definition edited by Andrew Howroyd, Jan 13 2025

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

cp's OEIS Frontend

A277741 Array read by antidiagonals: A(n,k) is the number of unsensed planar maps with n vertices and k faces, n >= 1, k >= 1.

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