A379431 -id:A379431 - 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

A379430 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 14, 23, 14, 3, 6, 42, 108, 108, 42, 6, 14, 140, 501, 761, 501, 140, 14, 34, 473, 2264, 4744, 4744, 2264, 473, 34, 95, 1670, 10087, 27768, 38495, 27768, 10087, 1670, 95, 280, 5969, 44310, 153668, 279698, 279698, 153668, 44310, 5969, 280

Offset: 1

Andrew Howroyd, Jan 13 2025

The planar maps considered are connected and may contain loops and parallel edges.

The number of edges is n + k - 2.

			Array begins:
=========================================================
n\k |  1    2     3      4      5      6      7     8 ...
----+----------------------------------------------------
  1 |  1    1     1      2      3      6     14    34 ...
  2 |  1    2     5     14     42    140    473  1670 ...
  3 |  1    5    23    108    501   2264  10087 44310 ...
  4 |  2   14   108    761   4744  27768 153668 ...
  5 |  3   42   501   4744  38495 279698 ...
  6 |  6  140  2264  27768 279698 ...
  7 | 14  473 10087 153668 ...
  8 | 34 1670 44310 ...
   ...
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,   14,    23,    14,     3;
   6,   42,   108,   108,    42,     6;
  14,  140,   501,   761,   501,   140,    14;
  34,  473,  2264,  4744,  4744,  2264,   473,   34;
  95, 1670, 10087, 27768, 38495, 27768, 10087, 1670, 95;
  ...

Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, pp. 1-10.

Antidiagonal sums are A006384.
Columns 1..2 are A002995, A380237.
Cf. A269920 (rooted), A277741 (unsensed), A379431 (achiral), A342061 (2-connected), A384964 (simple).

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

A380238 Number of achiral planar maps with n vertices and 2 faces.

Original entry on oeis.org

1, 2, 5, 12, 28, 68, 157, 372, 845, 1949, 4367, 9880, 21858, 48679, 106612, 234546, 509246, 1109352, 2391299, 5167423, 11070598, 23762557, 50641725, 108085708, 229303142, 487039228, 1029167119, 2176808877, 4583856878, 9660020146, 20279242545, 42599286814

Offset: 1

Andrew Howroyd, Jan 19 2025

nonn

Also by duality the number of achiral planar maps with n faces and 2 vertices.

The number of edges is n.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column 2 of A379431.

Cf. A380237 (sensed), A380239 (unsensed).

G1(n)={my(g=(1-sqrt(1-4*x^2 + O(x*x^n)))/(2*x^2)); ((1 + x/(1-x-x^2*g)^2)^2/(1 - x^2*g^2) - 1)/2}
G2(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); sum(k=1, n, my(m=1+k%2); -log(1 - x^k*c(m*k)^(2/m))*eulerphi(k)/k, O(x*x^n))}
seq(n)={Vec(G1(n)+G2(n))/2}

A380617 Triangle read by rows: T(n,k) is the number of achiral combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 8, 5, 2, 16, 28, 26, 12, 3, 53, 121, 128, 82, 28, 6, 206, 528, 686, 505, 239, 68, 10, 817, 2516, 3638, 3192, 1802, 686, 157, 20, 3620, 12302, 20250, 19976, 13268, 6078, 1876, 372, 35, 16361, 63643, 114669, 126876, 95422, 50954, 19346, 5100, 845, 70

Offset: 0

Andrew Howroyd, Jan 28 2025

By duality, also the number of achiral combinatorial maps with n edges and k faces.

			Triangle begins:
n\k |    1      2      3      4      5     6     7    8   9
----+-------------------------------------------------------
  0 |    1;
  1 |    1,     1;
  2 |    2,     2,     1;
  3 |    5,     8,     5,     2;
  4 |   16,    28,    26,    12,     3;
  5 |   53,   121,   128,    82,    28,    6;
  6 |  206,   528,   686,   505,   239,   68,   10;
  7 |  817,  2516,  3638,  3192,  1802,  686,  157,  20;
  8 | 3620, 12302, 20250, 19976, 13268, 6078, 1876, 372, 35;
  ...

Row sums are A170947.
Main diagonal is A001405(n-1).
Column 1 is A018191.
Cf. A379431 (planar), A380615 (sensed), A380616 (unsensed).

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A379430 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces, n >= 1, k >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A380238 Number of achiral planar maps with n vertices and 2 faces.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A380617 Triangle read by rows: T(n,k) is the number of achiral combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

Views

Author

Keywords

Comments

Examples

Crossrefs