A341856 -id:A341856 - cp's OEIS frontend

A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

1, 1, 3, 13, 68, 399, 2530, 16965, 118668, 857956, 6369883, 48336171, 373537388, 2931682810, 23317105140, 187606350645, 1524813969276, 12504654858828, 103367824774012, 860593023907540, 7211115497448720, 60776550501588855

Offset: 0

N. J. A. Sloane

Number of rooted loopless planar maps with n edges. E.g., there are a(2)=3 loopless planar maps with 2 edges: two rooted paths (.-.-.) and one digon (.=.). - Valery A. Liskovets, Sep 25 2003
Number of intervals (i.e., ordered pairs (x,y) such that x<=y) in the Tamari lattice (rotation lattice of binary trees) of size n (see Pallo and Chapoton references). - Ralf Stephan, May 08 2007, Jean Pallo (Jean.Pallo(AT)u-bourgogne.fr), Sep 11 2007
Number of rooted triangulations of type [n, 0] (see Brown paper eq (4.8)). - Michel Marcus, Jun 23 2013
Equivalently, number of rooted bridgeless planar maps with n edges. - Noam Zeilberger, Oct 06 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
Number of uniquely sorted permutations of [2n+1] that avoid the pattern 231. Also the number of uniquely sorted permutations of [2n+1] that avoid 132. - Colin Defant, Jun 13 2019
The sequence 1,3,13,68,... appears naturally in integral geometry, namely in the algebra of unitarily invariant valuations on complex space forms. - Andreas Bernig, Feb 02 2020

			G.f. = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 399*x^5 + 2530*x^6 + 16965*x^7 + ...

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
Handbook of Combinatorics, North-Holland '95, p. 891.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. T. Tutte, The enumerative theory of planar maps, in A Survey of Combinatorial Theory (J. N. Srivastava et al. eds.), pp. 437-448, North-Holland, Amsterdam, 1973.

T. D. Noe, Table of n, a(n) for n = 0..200
E. A. Bender and N. C. Wormald, The number of loopless planar maps, Discr. Math. 54:2 (1985), 235-237.
Andreas Bernig, Unitarily invariant valuations and Tutte's sequence, arXiv:2001.03372 [math.DG], 2020.
Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See pp. 10-11.
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
T. Daniel Brennan, Christian Ferko, and Savdeep Sethi, A Non-Abelian Analogue of DBI from $T \overline{T}$, arXiv:1912.12389 [hep-th], 2019. See also SciPost Phys. (2020) Vol. 8, 052.
Enrica Duchi and Corentin Henriet, A bijection between Tamari intervals and extended fighting fish, arXiv:2206.04375 [math.CO], 2022.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
Frédéric Chapoton, Sur le nombre d'intervalles dans les treillis de Tamari, arXiv:math/0602368 [math.CO], 2006.
Grégory Chatel, Vincent Pilaud, and Viviane Pons, The weak order on integer posets, arXiv:1701.07995 [math.CO], 2017.
Grégory Chatel and Viviane Pons, Counting smaller elements in the Tamari and m-Tamari lattices, arXiv preprint arXiv:1311.3922 [math.CO], 2013.
François David, A model of random surfaces with nontrivial critical behavior, Nuclear Physics B, v. 257 (1985), 543-576.
Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
Wenjie Fang, Planar triangulations, bridgeless planar maps and Tamari intervals, arXiv:1611.07922 [math.CO], 2016.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
C. Germain and J. Pallo, The number of coverings in four Catalan lattices, Intern. J. Computer Math., Vol. 61 (1996) pp. 19-28. (See p. 27.)
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
Zhaoxiang Li and Yanpei Liu, Chromatic sums of general maps on the sphere and the projective plane, Discr. Math. 307 (2007), 78-87.
Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).
K. A. Penson, K. Górska, A. Horzela, and G. H. E. Duchamp, Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution, arXiv:2209.06574 [math.CO], 2022.
Viviane Pons, Combinatorics of the Permutahedra, Associahedra, and Friends, arXiv:2310.12687 [math.CO], 2023. See p. 37.
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38. See Eq. 5.12.
William T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), no. 2, 105-126. MR0572468 (81j:05073).
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, eq. (6).
Noam Zeilberger, A sequent calculus for the Tamari order, arXiv:1701.02917 [cs.LO], 2017.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030 [math.LO], 2018-2019 (A revised version of a 2017 conference paper).
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. See also Part 2.

Row sums of A342981.
Column 0 of A146305 and A341856; Column 2 of A255918.
Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A000256, A006013, A027836, A069271, A263191, A290326.

[Binomial(4*n+1, n+1)-9*Binomial(4*n+1, n-1): n in [0..25]]; // Vincenzo Librandi, Nov 24 2016

A000260 := proc(n)
    2*(4*n+1)!/((n+1)!*(3*n+2)!) ;
end proc:

Table[Binomial[4n+1,n+1]-9*Binomial[4n+1,n-1],{n,0,25}] (* Harvey P. Dale, Aug 23 2011 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 3/4, 1, 5/4}, {4/3, 5/3, 2}, 256/27 x], {x, 0, n}]; (* Michael Somos, Dec 23 2014 *)
terms = 22; G[] = 0; Do[G[x] = 1+x*G[x]^4 + O[x]^terms, terms];
CoefficientList[(2-G[x])*G[x]^2, x] (* Jean-François Alcover, Jan 13 2018, after Mark van Hoeij *)

{a(n) = if( n<0, 0, 2 * (4*n + 1)! / ((n + 1)! * (3*n + 2)!))}; /* Michael Somos, Sep 07 2005 */

{a(n) = binomial( 4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2))}; /* Michael Somos, Mar 28 2012 */

def a(n):
    n = ZZ(n)
    return (4*n + 2).binomial(n + 1) // ((2*n + 1) * (3*n + 2))
# F. Chapoton, Aug 06 2015

a(n) = 2*(4*n+1)! / ((n+1)!*(3*n+2)!) = binomial(4*n+1, n+1) - 9*binomial(4*n+1, n-1).
G.f.: (2-g)*g^2 where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 10 2011
G.f.: hypergeom([1,1/2,3/4,5/4],[2,4/3,5/3],256*x/27) = 1 + 120*x/(Q(0)-120*x); Q(k) = 8*x*(2*k+1)*(4*k+3)*(4*k+5) + 3*(k+2)*(3*k+4)*(3*k+5) - 24*x*(k+2)*(2*k+3)*(3*k+4)*(3*k+5)*(4*k+7)*(4*k+9)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2011
a(n) = binomial(4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2)). - Michael Somos, Mar 28 2012
a(n) * (n+1) = A069271(n). - Michael Somos, Mar 28 2012
0 = F(a(n), a(n+1), ..., a(n+8)) for all n in Z where a(-1) = 3/4 and F() is a polynomial of degree 2 with integer coefficients and 29 monomials. - Michael Somos, Dec 23 2014
D-finite with recurrence: 3*(3*n+2)*(3*n+1)*(n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 21 2015
a(n) = Sum_{k=1..A000108(n)} k * A263191(n,k). - Alois P. Heinz, Nov 16 2015
a(n) ~ 2^(8*n+7/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n+5/2)). - Vaclav Kotesovec, Feb 26 2016
E.g.f.: 3F3(1/2,3/4,5/4; 4/3,5/3,2; 256*x/27). - Ilya Gutkovskiy, Feb 01 2017
From Gheorghe Coserea, Aug 17 2017: (Start)
G.f. y(x) satisfies:
0 = x^3*y^4 + 3*x^2*y^3 + x*(8*x+3)*y^2 - (20*x-1)*y + 16*x-1.
0 = x*(256*x - 27)*deriv(y,x) - 8*x^2*y^3 - 25*x*y^2 + 4*(24*x-11)*y + 44.
(End)
From Karol A. Penson, Apr 06 2022: (Start)
a(n) = Integral_{x=0...256/27} x^n*W(x), where
W(x) = (sqrt(2)/Pi)*(h1(x) - h2(x) + h3(x)) and
h1(x) = 3F2([-6/12,-2/12,  2/12], [ 3/12,  9/12], (27*x)/256)/((x/2)^(1/2));
h2(x) = 3F2([-3/12, 1/12,  5/12], [ 6/12, 15/12], (27*x)/256)/(x^(1/4));
h3(x) = 3F2([ 3/12, 7/12, 11/12], [18/12, 21/12], (27*x)/256)/(x^(-1/4)*32).
This integral representation is unique as the solution of n-th Hausdorff power moment of the function W. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 256/27. (End)
a(n) = (1/27^n) * Product_{1 <= i <= j <= 3*n} (3*i + j + 3)/(3*i + j - 1). Cf. A006013. - Peter Bala, Feb 21 2023

Edited by F. Chapoton, Feb 03 2011

A210664 Square array read by upwards antidiagonals: T(m,n) is the number of simple 3-connected triangulations of a closed region in the plane with m+3 given external edges and 3n+m internal edges, m>=0, n>=1.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 5, 6, 3, 1, 9, 20, 22, 12, 1, 14, 50, 85, 91, 52, 1, 20, 105, 254, 385, 408, 241, 1, 27, 196, 644, 1287, 1836, 1938, 1173, 1, 35, 336, 1448, 3696, 6630, 9120, 9614, 5929, 1, 44, 540, 2967, 9468, 20790, 34846, 46805, 49335, 30880

Offset: 0

N. J. A. Sloane, Mar 28 2012

A triangulation is simple if it contains no separating 3-cycle. There are n interior nodes and m+3 nodes on the boundary. - Andrew Howroyd, Feb 24 2021

			Array begins:
  1,  0,   1,    3,   12, ... (A000256)
  1,  2,   6,   22,   91, ...
  1,  5,  20,   85,  385, ...
  1,  9,  50,  254, 1287, ...
  1, 14, 105,  644, 3696, ...
  1, 20, 196, 1448, 9468, ...
  ...
From _Andrew Howroyd_, Feb 24 2021: (Start)
The array transposed for comparability with A341856 begins:
==================================================
n\m |   0    1    2     3      4      5      6
----+---------------------------------------------
  1 |   1    1    1     1      1      1      1 ...
  2 |   0    2    5     9     14     20     27 ...
  3 |   1    6   20    50    105    196    336 ...
  4 |   3   22   85   254    644   1448   2967 ...
  5 |  12   91  385  1287   3696   9468  22131 ...
  6 |  52  408 1836  6630  20790  58564 151146 ...
  7 | 241 1938 9120 34846 116641 353056 983664 ...
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1325
P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138.
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.

Rows m=0..3 are A000256, A000139, A341920, A341921.
Columns are A000012, A000096, A002415, A004305.
Antidiagonal sums give A341922.
Cf. A341856.

\\ here H is A000256 as g.f., U(n,m) is A341856 for m > 0.
H(n)={my(g=1+serreverse(x/(1+x)^4 + O(x*x^n) )); 2 - sqrt(serreverse(x*(2-g)^2*g^4)/x)}
U(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)!))}
R(N, m)={my(g=2-H(N)); Vec(if(m==0, 1-g, g^(m+1)*subst(O(x*x^N) + sum(n=1, N, U(n,m)*x^n), x, x*g^2)))}
M(m, n=m)={Mat(vectorv(m+1, i, R(n,i-1)))}
M(7) \\ Andrew Howroyd, Feb 23 2021

From Andrew Howroyd, Feb 24 2021: (Start)
G.f. of row m > 0: R(x) satisfies g(x^2)^(m+1)*R(x*g(x^2)) = B(x^2) where g(x) is the g.f. of column 0 of A341856 and B(x) is the g.f. of column m of A341856.
G.f. of row m > 0: h(x)^(m+1)*B(x*h(x)^2) where 2-h(x) is the g.f. of A000256 and B(x) is the g.f. of column m of A341856.
(End)

Terms a(21) and beyond from Andrew Howroyd, Feb 23 2021

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.

A342053 Array read by antidiagonals: T(n,k) is the number of unrooted 3-connected triangulations of a disk with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 2, 8, 16, 1, 3, 12, 38, 78, 1, 3, 20, 73, 219, 457, 1, 4, 27, 140, 503, 1404, 2938, 1, 4, 39, 235, 1089, 3661, 9714, 20118, 1, 5, 51, 392, 2149, 8796, 27715, 70454, 144113, 1, 5, 68, 610, 4050, 19419, 72204, 214664, 527235, 1065328

Offset: 1

Andrew Howroyd, Feb 26 2021

For k >= 4, T(n,k) is the number of polyhedra with n+k vertices whose faces are all triangular, except one which is k-gonal.

The initial terms of this sequence can also be computed using the tool "plantri", in particular the command "./plantri -u -v -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 |     4     8     12     20      27      39 ...
  4 |    16    38     73    140     235     392 ...
  5 |    78   219    503   1089    2149    4050 ...
  6 |   457  1404   3661   8796   19419   40485 ....
  7 |  2938  9714  27715  72204  173779  393123 ...
  8 | 20118 70454 214664 596906 1538221 3723976 ...
  ...

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)
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
Andrew Howroyd, PARI Program

Columns k=3..6 are A002713, A058786(n+4), A342054, A342055.
Antidiagonal sums are A342056.
Cf. A169808 (2-connected), A341856 (rooted), A341923 (oriented).

A342053Array(8,6) \\ See links for program.

A341917 Number of strong triangulations of a fixed pentagon with n interior nodes.

Original entry on oeis.org

1, 10, 80, 610, 4625, 35322, 272800, 2132310, 16861455, 134781350, 1088074064, 8863227100, 72789560025, 602232150810, 5016320750400, 42041168169712, 354326295908580, 3001705818071400, 25549873091193600, 218426240780417250, 1874878554813727815, 16153461053030605590

Offset: 1

Andrew Howroyd, Feb 23 2021

nonn

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

Column k=2 of A341856.

Cf. A341853.

a(n)={60*(13*n+9)*(4*n+5)!/((3*n+9)!*(n-1)!)}

a(n) = 60*(13*n+9)*(4*n+5)!/((3*n+9)!*(n-1)!).

A341918 Number of strong triangulations of a fixed hexagon with n interior nodes.

Original entry on oeis.org

1, 15, 155, 1410, 12165, 102548, 856205, 7128420, 59386655, 495973610, 4156326090, 34965061950, 295326872685, 2504471904060, 21322339850120, 182220932483936, 1562911922345700, 13451387178653575, 116149864521206475, 1006036370555080950, 8739349976276626707, 76128234963921173340

Offset: 1

Andrew Howroyd, Feb 23 2021

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

Column k=3 of A341856.

a(n) = {120*(67*n^2 + 138*n + 68)*(4*n+7)!/((3*n+12)!*(n-1)!)}

a(n) = 120*(67*n^2 + 138*n + 68)*(4*n+7)!/((3*n+12)!*(n-1)!).

a(n) ~ 335*2^(35/2+8*n)*3^(-23/2-3*n)/(n^(5/2)*sqrt(Pi)). - Stefano Spezia, Aug 25 2025

A242136 Number of strong triangulations of a fixed square with n interior vertices.

Original entry on oeis.org

0, 1, 6, 36, 228, 1518, 10530, 75516, 556512, 4194801, 32224114, 251565996, 1991331720, 15953808780, 129171585690, 1055640440268, 8698890336576, 72215877581844, 603532770013080, 5074488683389840

Offset: 0

David Callan, Aug 15 2014

nonn

A strong triangulation is one in which no interior edge joins two vertices of the square (see W. G. Brown reference).

If the restriction "strong" is dropped, the counting sequence is A197271 (shifted left).

			The 6 triangulations for n=2 are as follows. Four have a central vertex joined to all 4 vertices of the square creating 4 triangular regions, one of which contains the second interior vertex. In these 4 cases, the central vertex has degree 5, the other interior  vertex has degree 3. In the other 2 triangulations, both interior vertices have degree 4, an opposite pair a, c of vertices of the square both have degree 3 (so 1 interior edge), and the other 2 opposite vertices have degree 4.

William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14, Issue 4, (1964) 746-768.
William T. Tutte, A census of planar triangulations (Eq. 5.12), Canad. J. Math. 14 (1962), 21-38.

Column k=1 of A341856.

Cf. A000260 for triangulations of a triangle.

A242136:=n->24*binomial(4*n+3,n-1)/((3*n+5)*(n+2)): seq(A242136(n), n=0..30); # Wesley Ivan Hurt, Aug 16 2014

Table[24 Binomial[4n+3,n-1]/((3n+5)(n+2)), {n, 0, 15}]

a(n) = 72 * (4*n+3)!/((3*n+6)!*(n-1)!) = 24 * binomial(4*n+3,n-1)/((3*n+5)*(n+2)) = binomial(4*n+3,n-1) - 5 * binomial(4*n+3,n-2) + 6 * binomial(4*n+3,n-3).

cp's OEIS Frontend

A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A210664 Square array read by upwards antidiagonals: T(m,n) is the number of simple 3-connected triangulations of a closed region in the plane with m+3 given external edges and 3n+m internal edges, m>=0, n>=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A342053 Array read by antidiagonals: T(n,k) is the number of unrooted 3-connected triangulations of a disk with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A341917 Number of strong triangulations of a fixed pentagon with n interior nodes.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A341918 Number of strong triangulations of a fixed hexagon with n interior nodes.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A242136 Number of strong triangulations of a fixed square with n interior vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica