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

1,5

Comments

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)

References

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

Links

Table of n, a(n) for n=1..55.

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.

Formula

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

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

Example

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;
  ...

Crossrefs

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).

Sequence in context: A228815 A241555 A379431 * A379430 A241138 A241349

Adjacent sequences: A277738 A277739 A277740 * A277742 A277743 A277744

Keyword

nonn,tabl,changed

Author

N. J. A. Sloane, Nov 07 2016

Extensions

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

Status

approved