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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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

Examples

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

References

• 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.

Links

• 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.

Crossrefs

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.

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Sage

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

Formula

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

Extensions

Edited by F. Chapoton, Feb 03 2011