A035309 Triangle read by rows giving number of ways to glue sides of a 2n-gon so as to produce a surface of genus g.

1, 1, 2, 1, 5, 10, 14, 70, 21, 42, 420, 483, 132, 2310, 6468, 1485, 429, 12012, 66066, 56628, 1430, 60060, 570570, 1169740, 225225, 4862, 291720, 4390386, 17454580, 12317877, 16796, 1385670, 31039008, 211083730, 351683046, 59520825, 58786, 6466460, 205633428, 2198596400, 7034538511, 4304016990

Offset: 0

Author

Dylan Thurston

Keywords

• nonn
• tabf
• nice

Comments

Row n contains floor((n+2)/2) terms.

a(n,g) is also the number of unicellular (i.e., 1-faced) rooted maps of genus g with n edges. #(vertices) = n - 2g + 1. Dually, this is the number of 1-vertex maps. Catalan(n)=A000108(n) divides a(n,g)2^g.

From Akhmedov and Shakirov's abstract: By pairwise gluing of sides of a polygon, one produces two-dimensional surfaces with handles and boundaries. We give the number N_{g,L}(n_1, n_2, ..., n_L) of different ways to produce a surface of given genus g with L polygonal boundaries with given numbers of sides n_1, n_2, >..., n_L. Using combinatorial relations between graphs on real two-dimensional surfaces, we derive recursive relations between N_{g,L}. We show that Harer-Zagier numbers appear as a particular case of N_{g,L} and derive a new explicit expression for them. - Jonathan Vos Post, Dec 18 2007

Examples

			Triangle starts:
n\g    [0]        [1]        [2]        [3]        [4]        [5]
[0]    1;
[1]    1;
[2]    2;         1;
[3]    5,         10;
[4]    14,        70,        21;
[5]    42,        420,       483;
[6]    132,       2310,      6468,      1485;
[7]    429,       12012,     66066,     56628;
[8]    1430,      60060,     570570,    1169740,   225225;
[9]    4862,      291720,    4390386,   17454580,  12317877;
[10]   16796,     1385670,   31039008,  211083730, 351683046, 59520825;
[11]   ...
		

Links

• Gheorghe Coserea, Rows n = 0..200, flattened
• E. T. Akhmedov and Sh. Shakirov, Gluing of Surfaces with Polygonal Boundaries, arXiv:0712.2448 [math.CO], 2007-2008, see p. 1.
• Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.
• Ricky X. F. Chen and Christian M. Reidys, A Combinatorial Identity Concerning Plane Colored Trees and its Applications, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.7.
• Benoit Collins, Ion Nechita, and Deping Ye, The absolute positive partial transpose property for random induced states, Random Matrices: Theory Appl. 01, 1250002 (2012); arXiv:1108.1935 [math-ph], 2011.
• I. P. Goulden and A. Nica, A direct bijection for the Harer-Zagier formula, J. Comb. Theory, A, 111, No. 2 (2005), 224-238.
• J. L. Harer and D. B. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math., 85, No.3 (1986), 457-486.
• S. Lando and A. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, 141, Springer, 2004, p. 157.
• B. Lass, Démonstration combinatoire de la formule de Harer-Zagier, C. R. Acad. Sci. Paris, Serie I, 333, No.3 (2001), 155-160.
• A. Mironov, A. Morozov, A. Popolitov, and Sh. Shakirov, Summing up perturbation series around superintegrable point, arXiv:2401.14392 [hep-th], 2024.
• T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Theory B 13 (1972), 192-218 (Tab. 1).
• Nikolai Wyderka and Andreas Ketterer, Probing the geometry of correlation matrices with randomized measurements, arXiv:2211.09610 [quant-ph], 2022.
• Liang Zhao and Fengyao Yan, Note on Total Positivity for a Class of Recursive Matrices, Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.5.
• Jian Zhou, Hermitian One-Matrix Model and KP Hierarchy, arXiv:1809.07951 [math-ph], 2018.

Crossrefs

Row sums give A001147(n).

Columns g=0-2 give: A000108, A002802, A006298.

The last entries in the even rows give A035319.

Cf. A270406, A270790, A270791.

Programs

Mathematica

a[n_, g_] := (2n)!/(n+1)!/(n-2g)! Coefficient[Series[(x/2/Tanh[x/2])^(n+1), {x, 0, n}], x, 2g]; Flatten[DeleteCases[#, 0]& /@ Table[a[n, g], {n, 0, 11}, {g, 0, n}]] (* Jean-François Alcover, Aug 30 2011, after E. T. Akhmedov & Sh. Shakirov *)

PARI

N = 10; F = 1; gmax(n) = n\2;
Q = matrix(N + 1, N + 1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, length(Q)-1, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric('x + O('x^(F+1)));
concat(vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F))))
\\ Gheorghe Coserea, Mar 16 2016

Formula

Let c be the number of cycles that appear in product of a (2n)-cycle and a product of n disjoint transpositions; genus is g = (n-c+1)/2.

The Harer-Zagier formula: 1 + 2*Sum_{g>=0} Sum_{n>=2*g} a(n,g) * x^(n+1) * y^(n-2*g+1) / (2*n-1)!! = (1+x/(1-x))^y.

Equivalently, for n >= 0, Sum_{g=0..floor(n/2)} a(n,g)*y^(n-2*g+1) = (2*n-1)!! * Sum_{k=0..n} 2^k * C(n,k) * C(y,k+1).

(n+2)*a(n+1,g) = (4*n+2)*a(n,g) + (4*n^3-n)*a(n-1,g-1) for n,g > 0, a(0,0)=1 and a(0,g)=0 for g > 0.

The g.f. for column g > 0 is x^(2*g) * A270790(g) * P_g(x) / (1-4*x)^(3*g-1/2), where P_g(x) is the polynomial associated with row g of the triangle A270791. - Gheorghe Coserea, Apr 17 2016

Extensions

More terms and additional comments and references from Valery A. Liskovets, Apr 13 2006

Offset corrected by Gheorghe Coserea, Mar 17 2016