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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. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

LINKS

Gheorghe Coserea, Rows n = 0..200, flattened

Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

E. T. Akhmedov and Sh. Shakirov, Gluing of Surfaces with Polygonal Boundaries, arXiv:0712.2448 [math.CO], 2007-2008, see p. 1.

Benoit Collins, Ion Nechita, 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.

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Theory B 13 (1972), 192-218 (Tab.1).

FORMULA

Let c be 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(0<=g<=floor(n/2), a(n,g)*y^(n-2*g+1) ) = (2*n-1)!! * Sum(0<=k<=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

EXAMPLE

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]   ...

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 *)

PROG

(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

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.

Sequence in context: A178627 A247368 A019098 * A174218 A226308 A226309

Adjacent sequences:  A035306 A035307 A035308 * A035310 A035311 A035312

KEYWORD

nonn,tabf,nice

AUTHOR

Dylan Thurston

EXTENSIONS

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

Offset corrected by Gheorghe Coserea, Mar 17 2016

STATUS

approved

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Last modified June 30 08:05 EDT 2016. Contains 274307 sequences.