A342053 -id:A342053 - cp's OEIS frontend

A341856 Array read by antidiagonals: T(n,k) is the number of rooted strong triangulations of a disk with n interior nodes and 3+k nodes on the boundary.

1, 0, 1, 0, 1, 3, 0, 1, 6, 13, 0, 1, 10, 36, 68, 0, 1, 15, 80, 228, 399, 0, 1, 21, 155, 610, 1518, 2530, 0, 1, 28, 273, 1410, 4625, 10530, 16965, 0, 1, 36, 448, 2933, 12165, 35322, 75516, 118668, 0, 1, 45, 696, 5628, 28707, 102548, 272800, 556512, 857956

Offset: 0

Andrew Howroyd, Feb 23 2021

A strong triangulation is one in which no interior edge joins two nodes on the boundary. Except for the single triangle which is enumerated by T(0,0) these are the 3-connected triangulations.

			Array begins:
=======================================================
n\k |    0     1     2      3      4      5       6
----+--------------------------------------------------
  0 |    1     0     0      0      0      0       0 ...
  1 |    1     1     1      1      1      1       1 ...
  2 |    3     6    10     15     21     28      36 ...
  3 |   13    36    80    155    273    448     696 ...
  4 |   68   228   610   1410   2933   5628   10128 ...
  5 |  399  1518  4625  12165  28707  62230  125928 ...
  6 | 2530 10530 35322 102548 267162 638624 1422204 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.

Columns k=0..3 are A000260, A242136, A341917, A341918.
Antidiagonal sums give A341919.
Cf. A146305 (not necessarily strong triangulations), A210664, A341923, A342053.

T(n,m)=if(m==0, 2*(4*n+1)!/((3*n+2)!*(n+1)!), (3*(m+2)!*(m-1)!/(3*n+3*m+3)!)*sum(j=0, min(m,n-1), (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!)))

T(n,0) = A000260(n) = 2*(4*n+1)!/((3*n+2)!*(n+1)!).

T(n,m) = (3*(m+2)!*(m-1)!/(3*n+3*m+3)!) * Sum_{j=0..min(m,n-1)} (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!) for m > 0.

A341923 Array read by antidiagonals: T(n,k) is the number of 3-connected triangulations of a disk up to orientation-preserving isomorphisms with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 1, 2, 10, 24, 1, 3, 16, 60, 133, 1, 3, 28, 122, 386, 846, 1, 4, 39, 242, 925, 2652, 5661, 1, 4, 58, 419, 2039, 7066, 18914, 39556, 1, 5, 78, 711, 4101, 17138, 54560, 139264, 286000, 1, 5, 106, 1128, 7801, 38166, 142802, 426462, 1048947, 2123329

Offset: 1

Andrew Howroyd, Feb 26 2021

The initial terms of this sequence can also be computed using the tool "plantri", in particular the command "./plantri -u -v -o -P [n]" will compute values for a diagonal.

			Array begins:
=====================================================
n\k |     3      4      5       6       7       8
----+------------------------------------------------
  1 |     1      1      1       1       1       1 ...
  2 |     1      2      2       3       3       4 ...
  3 |     5     10     16      28      39      58 ...
  4 |    24     60    122     242     419     711 ...
  5 |   133    386    925    2039    4101    7801 ...
  6 |   846   2652   7066   17138   38166   79908 ...
  7 |  5661  18914  54560  142802  345099  782210 ...
  8 | 39556 139264 426462 1188412 3067938 7433635 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
G. Brinkmann and B. McKay, Plantri (program for generation of certain types of planar graph)

Columns k=3..6 are A002709, A341924, A341925, A341926.
Antidiagonal sums are A342052.
Cf. A262586 (2-connected), A341856 (rooted), A342053 (unrooted).

A341923Array(8,6) \\ See links in A342053 for program file.

A058786 Number of n-hedra with 2n-5 vertices or 3n-7 edges (the vertices of these are all of degree 3, except one which is of degree 4). Alternatively, the number of polyhedra with n vertices whose faces are all triangular, except one which is tetragonal.

Original entry on oeis.org

1, 2, 8, 38, 219, 1404, 9714, 70454, 527235, 4037671, 31477887, 249026400, 1994599707, 16147744792, 131959532817, 1087376999834, 9027039627035, 75441790558926, 634311771606750, 5362639252793358, 45565021714371644, 388937603694422120, 3333984869758146814

Offset: 5

Gerard P. Michon, Nov 29 2000

			a(5)=1 because the square pyramid is the only pentahedron with 5=2*5-5 vertices (or 8=3*5-7 edges). Alternatively, a(5)=1 because the square pyramid is the only polyhedron with 5 vertices whose faces are all triangles with only one tetragonal exception.

Andrew Howroyd, Table of n, a(n) for n = 5..500
CombOS - Combinatorial Object Server, generate planar graphs
G. P. Michon, Counting Polyhedra

Column k=4 of A342053.

Cf. A000109, A002856, A000944, A002840, A058787, A058788, A049337.

A342053ColSeq(25,4) \\ See links in A342053. - Andrew Howroyd, Feb 28 2021

Terms a(19) and beyond from Andrew Howroyd, Feb 27 2021

A342056 Number of unrooted 3-connected triangulations of a disk with n nodes.

Original entry on oeis.org

1, 2, 7, 27, 132, 773, 5017, 34861, 253676, 1903584, 14616442, 114254053, 906266345, 7277665889, 59066524810, 483864411124, 3996427278475, 33250623548406, 278472800696431, 2346089674759665, 19872284655990058, 169155659546689252, 1446375853153588731, 12418636483734071261, 107034910337046043232

Offset: 4

Andrew Howroyd, Feb 26 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 4..100
G. Brinkmann and B. D. McKay, Fast generation of planar graphs (expanded edition), Table 27, Column d_1.
The House of Graphs, Planar graphs

Antidiagonal sums of A342053.

Cf. A342052.

A342053AntidiagonalSums(25) \\ See links in A342053 for program.

A342052 Number of 3-connected triangulations of a disk with n nodes up to orientation-preserving isomorphisms.

Original entry on oeis.org

1, 2, 8, 37, 213, 1386, 9524, 68057, 501858, 3788747, 29170667, 228295618, 1811802818, 14552804492, 118124257451, 967698049455, 7992746427963, 66500865364037, 556944249243331, 4692174542007030, 39744552170122779, 338311257783873501, 2892751486485359650

Offset: 4

Andrew Howroyd, Feb 26 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 4..100
G. Brinkmann and B. D. McKay, Fast generation of planar graphs (expanded edition), Table 27, Column d'1.

Antidiagonal sums of A341923.

Cf. A342053, A342056.

A341923AntidiagonalSums(25) \\ See links in A342053 for program.

A342054 Number of unrooted 3-connected triangulations of a pentagon with n interior nodes.

Original entry on oeis.org

1, 2, 12, 73, 503, 3661, 27715, 214664, 1691049, 13494718, 108864742, 886520081, 7279644889, 60225617740, 501640531350, 4204146605148, 35432735188076, 300170956721542, 2554988646383521, 21842628856370472, 187487872615533629, 1615346166853635465, 13965841907774754081

Offset: 1

Andrew Howroyd, Feb 26 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column k=5 of A342053.

Cf. A341925.

A342055 Number of unrooted 3-connected triangulations of a hexagon with n interior nodes.

Original entry on oeis.org

1, 3, 20, 140, 1089, 8796, 72204, 596906, 4958736, 41365110, 346477770, 2914165157, 24612003577, 208711039484, 1776879432103, 15185140895506, 130242884479498, 1120949733518971, 9679158242220797, 83836374509942511, 728279201686023394, 6344019713842496995

Offset: 1

Andrew Howroyd, Feb 26 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column k=6 of A342053.

Cf. A341926.

cp's OEIS Frontend

A341856 Array read by antidiagonals: T(n,k) is the number of rooted strong triangulations of a disk with n interior nodes and 3+k nodes on the boundary.

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A341923 Array read by antidiagonals: T(n,k) is the number of 3-connected triangulations of a disk up to orientation-preserving isomorphisms with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.

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A058786 Number of n-hedra with 2n-5 vertices or 3n-7 edges (the vertices of these are all of degree 3, except one which is of degree 4). Alternatively, the number of polyhedra with n vertices whose faces are all triangular, except one which is tetragonal.

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A342056 Number of unrooted 3-connected triangulations of a disk with n nodes.

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A342052 Number of 3-connected triangulations of a disk with n nodes up to orientation-preserving isomorphisms.

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A342054 Number of unrooted 3-connected triangulations of a pentagon with n interior nodes.

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A342055 Number of unrooted 3-connected triangulations of a hexagon with n interior nodes.

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