A239893 -id:A239893 - cp's OEIS frontend

A212438 Irregular triangle read by rows: T(n,k) is the number of polyhedra with n faces and k vertices (n >= 4, k=4..2n-4).

1, 0, 1, 1, 0, 1, 2, 2, 2, 0, 0, 2, 8, 11, 8, 5, 0, 0, 2, 11, 42, 74, 76, 38, 14, 0, 0, 0, 8, 74, 296, 633, 768, 558, 219, 50, 0, 0, 0, 5, 76, 633, 2635, 6134, 8822, 7916, 4442, 1404, 233, 0, 0, 0, 0, 38, 768, 6134, 25626, 64439, 104213, 112082, 79773, 36528, 9714, 1249

Offset: 4

N. J. A. Sloane, May 16 2012

Because of duality, T(n,k) = T(k,n). - Ivan Neretin, May 25 2016

The number of edges is n+k-2. - Andrew Howroyd, Mar 27 2021

			Triangle begins:
1
0 1 1
0 1 2  2  2
0 0 2  8 11   8    5
0 0 2 11 42  74   76   38   14
0 0 0  8 74 296  633  768  558  219   50
0 0 0  5 76 633 2635 6134 8822 7916 4442 1404 233
...

Andrew Howroyd, Table of n, a(n) for n = 4..199 (rows 4..17)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Table 9-11.
G. P. Michon, Counting Polyhedra

A049337, A058787, A212438 are all versions of the same triangle.
Row sums (the same as column sums) are A000944.
Main diagonal is A002856.
Cf. A002840 (by edges), A239893.

Terms a(53) and beyond from Andrew Howroyd, Mar 27 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).

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).

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).

A005645 Number of sensed 3-connected planar maps with n edges.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 15, 32, 89, 266, 797, 2496, 8012, 26028, 85888, 286608, 965216, 3278776, 11221548, 38665192, 134050521, 467382224, 1638080277, 5768886048, 20407622631, 72494277840, 258527335373, 925322077852, 3323258053528, 11973883092034, 43273374700200, 156836969693756, 569967330200576, 2076647113454878, 7584534277720818, 27764845224462192, 101862027752012402, 374484866509396780, 1379489908513460150

Offset: 6

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 6..50
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices
T. R. S. Walsh, Counting nonisomorphic three-connected planar maps, J. Combin. Theory Ser. B 32 (1982), no. 1, 33-44.

Cf. A002840 (unsensed), A239893.

a(n) = Sum_{k=4..n-2} A239893(k, n+2-k). - Andrew Howroyd, Mar 27 2021

More terms and b-file added by N. J. A. Sloane, May 08 2012

A342057 Number of polyhedra with n faces and n vertices up to orientation preserving isomorphisms.

Original entry on oeis.org

1, 1, 3, 11, 69, 541, 5096, 50586, 534292, 5865150, 66582243, 776705379, 9274453627, 112984297173

Offset: 4

Andrew Howroyd, Mar 27 2021

Main diagonal of A239893.

Cf. A002856.

cp's OEIS Frontend

A212438 Irregular triangle read by rows: T(n,k) is the number of polyhedra with n faces and k vertices (n >= 4, k=4..2n-4).

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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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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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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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A005645 Number of sensed 3-connected planar maps with n edges.

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A342057 Number of polyhedra with n faces and n vertices up to orientation preserving isomorphisms.

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