A342059 -id:A342059 - cp's OEIS frontend

A342060 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, n >= 3, k=2..2*n-4.

1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 13, 21, 16, 5, 2, 1, 4, 29, 94, 183, 154, 76, 18, 5, 1, 6, 59, 328, 1146, 2114, 2144, 1246, 447, 88, 14, 1, 7, 104, 915, 5046, 16009, 30183, 33719, 23749, 10585, 3017, 489, 50, 1, 9, 181, 2239, 17876, 85550, 254831, 478913, 581324, 468388, 255156, 93028, 22077, 3071, 233

Offset: 3

Andrew Howroyd, Mar 27 2021

Equivalently, T(n,k) is the number of unsensed 2-connected planar maps with n vertices and k faces.
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,  4,   2,    1;
  1, 3, 13,  21,   16,    5,    2;
  1, 4, 29,  94,  183,  154,   76,   18,   5;
  1, 6, 59, 328, 1146, 2114, 2144, 1246, 447, 88, 14;
  ...

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 A034889.

Cf. A006407 (by edges), A212438 (3-connected), A342059.

T(n,2) = 1.

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

A384964 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces up to orientation preserving isomorphisms, n >= 1, k=1..max(1,2*n-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 8, 8, 6, 2, 1, 6, 29, 60, 73, 52, 25, 6, 2, 14, 113, 388, 768, 903, 728, 379, 136, 26, 6, 34, 444, 2303, 6584, 11782, 14321, 12113, 7298, 3048, 872, 147, 17, 95, 1763, 12650, 49806, 123547, 210314, 255884, 228807, 150929, 73428, 25536, 6142, 892, 73

Offset: 1

Andrew Howroyd, Jun 13 2025

Equivalently, T(n,k) is the number of sensed simple planar maps with n vertices and k faces.
The number of edges is n+k-2.
Terms of this sequence can be computed using the tool "plantri". The expanded reference gives rows 1..14 of this table.

			Triangle begins:
   1;
   1;
   1,   1,
   2,   2,    1,    1,
   3,   8,    8,    6,     2,     1,
   6,  29,   60,   73,    52,    25,     6,    2,
  14, 113,  388,  768,   903,   728,   379,  136,   26,   6,
  34, 444, 2303, 6584, 11782, 14321, 12113, 7298, 3048, 872, 147, 17;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..158 (rows 1..14)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Tables 19-22.

Row sums are A384965.
Antidiagonal sums are A006394.
Columns 1..2 are A002995, A384966.
Cf. A379430 (not necessarily simple), A342059 (2-connected), A239893 (3-connected), A384963 (unsensed).

A006406 Number of sensed 2-connected simple planar maps with n edges.

Original entry on oeis.org

1, 1, 2, 4, 9, 24, 81, 274, 1071, 4357, 18416, 80040, 356109, 1610910, 7399114

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 up to orientation preserving isomorphisms. - 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. A006402, A006404, A006407 (unsensed case), A342059, A379437 (rooted).

a(n) = Sum_{k=3..n} A342059(k, n+2-k). - Andrew Howroyd, Mar 27 2021

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

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

A342061 Triangle read by rows: T(n,k) is the number of sensed 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, 8, 3, 1, 1, 4, 16, 16, 4, 1, 1, 5, 38, 63, 38, 5, 1, 1, 7, 72, 218, 218, 72, 7, 1, 1, 8, 134, 622, 1075, 622, 134, 8, 1, 1, 10, 224, 1600, 4214, 4214, 1600, 224, 10, 1, 1, 12, 375, 3703, 14381, 22222, 14381, 3703, 375, 12, 1

Offset: 2

Andrew Howroyd, Mar 30 2021

The number of faces is n + 2 - k.

			Triangle begins:
  1;
  1, 1;
  1, 1,   1;
  1, 2,   2,   1;
  1, 3,   8,   3,    1;
  1, 4,  16,  16,    4,   1;
  1, 5,  38,  63,   38,   5,   1;
  1, 7,  72, 218,  218,  72,   7, 1;
  1, 8, 134, 622, 1075, 622, 134, 8, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)
Timothy R. Walsh, Efficient enumeration of sensed planar maps, Discrete Math. 293 (2005), no. 1-3, 263--289. MR2136069 (2006b:05062).

Column k=3 is A001399(n-3).
Row sums are A006402.
Cf. A082680 (rooted), A239893, A342059.

\\ See section 4 of Walsh reference.
T(n)={
  my(B=matrix(n, n, i, j, if(i+j <= n+1, (2*i+j-2)!*(2*j+i-2)!/(i!*j!*(2*i-1)!*(2*j-1)!))));
  my(C(i,j)=((i+j-1)*(i+1)*(j+1)/(2*(2*i+j-1)*(2*j+i-1)))*B[(i+1)/2,(j+1)/2]);
  my(D(i,j)=((j+1)/2)*B[i/2, (j+1)/2]);
  my(E(i,j)=((i-1)*(j-1) + 2*(i+j)*(i+j-1))*B[i,j]);
  my(F(i,j)=if(!i, j==1, ((i+j)*(6*j+2*i-5)*j*(2*i+j-1)/(2*(2*i+1)*(2*j+i-2)))*B[i,j]) + if(j-1, binomial(i+2,2)*B[i+1,j-1]));
  vector(n, n, vector(n, i, my(j=n+1-i); B[i,j]
    + (i+j)*if(i%2, if(j%2, C(i,j), D(j,i)), if(j%2, D(i,j)))
    + sumdiv(i+j, d, if(d>1, eulerphi(d)*( if(i%d==0, E(i/d, j/d) ) + if(i%d==1, F((i-1)/d, (j+1)/d)) + if(j%d==1, F((j-1)/d, (i+1)/d)) )))
   )/(2*n+2));
}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

T(n,k) = T(n, n+2-k).

A342058 Number of embeddings on the sphere of 2-connected planar graphs with n nodes up to orientation preserving isomorphisms.

Original entry on oeis.org

1, 3, 11, 82, 942, 14162, 242110, 4492455, 88181555, 1808322585, 38406536120, 839691224953, 18810933303213

Offset: 3

Andrew Howroyd, Mar 27 2021

Number of sensed 2-connected planar maps with n vertices. Multiple edges and loops are not allowed.

Row sums of A342059.

Cf. A034889 (unsensed case).

cp's OEIS Frontend

A342060 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, n >= 3, k=2..2*n-4.

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A384964 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces up to orientation preserving isomorphisms, n >= 1, k=1..max(1,2*n-4).

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A006406 Number of sensed 2-connected simple planar maps with n edges.

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A342061 Triangle read by rows: T(n,k) is the number of sensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.

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A342058 Number of embeddings on the sphere of 2-connected planar graphs with n nodes up to orientation preserving isomorphisms.

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