A267180 -id:A267180 - cp's OEIS frontend

A380235 Triangle read by rows: T(n,k) is the number of n edge non-orientable genus k maps.

1, 4, 2, 19, 16, 8, 106, 137, 128, 47, 709, 1254, 1890, 1372, 473, 5356, 12597, 27036, 31007, 22556, 7190, 44558, 133518, 379491, 611322, 704066, 469632, 144904, 397146, 1464725, 5229092, 11017122, 17691240, 18521632, 11990766, 3534490, 3716039, 16373700, 70805740, 186044902, 387965547, 563764626, 571333104, 352456980, 100895667

Offset: 1

Andrew Howroyd, Jan 17 2025

			Triangle begins:
      1;
      4,      2;
     19,     16,      8;
    106,    137,    128,     47;
    709,   1254,   1890,   1372,    473;
   5356,  12597,  27036,  31007,  22556,   7190;
  44558, 133518, 379491, 611322, 704066, 469632, 144904;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..105 (rows 1..14)
Evgeniy Krasko and Alexander Omelchenko, Enumeration of Unsensed Orientable Maps on Surfaces of a Given Genus, arXiv:1712.10139 [math.CO], 2017, see Appendix Tables 5-7.
Evgeniy Krasko, Maps on non-orientable surfaces, 2017 (data file).

Main diagonal is A380236.
Columns k=1..10 are A348802, A348803, A348804, A348805, A348806, A348807, A348808, A348809, A348810, A348811.
Cf. A267180 (rooted), A379439 (on orientable surface).

A238396 Triangle T(n,k) read by rows: T(n,k) is the number of rooted genus-k maps with n edges, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 2, 0, 9, 1, 0, 54, 20, 0, 0, 378, 307, 21, 0, 0, 2916, 4280, 966, 0, 0, 0, 24057, 56914, 27954, 1485, 0, 0, 0, 208494, 736568, 650076, 113256, 0, 0, 0, 0, 1876446, 9370183, 13271982, 5008230, 225225, 0, 0, 0, 0, 17399772, 117822512, 248371380, 167808024, 24635754, 0, 0, 0, 0, 0, 165297834, 1469283166, 4366441128, 4721384790, 1495900107, 59520825, 0

Offset: 0

Joerg Arndt, Feb 26 2014

			Triangle starts:
00: 1,
01: 2, 0,
02: 9, 1, 0,
03: 54, 20, 0, 0,
04: 378, 307, 21, 0, 0,
05: 2916, 4280, 966, 0, 0, 0,
06: 24057, 56914, 27954, 1485, 0, 0, 0,
07: 208494, 736568, 650076, 113256, 0, 0, 0, 0,
08: 1876446, 9370183, 13271982, 5008230, 225225, 0, 0, 0, 0,
09: 17399772, 117822512, 248371380, 167808024, 24635754, 0, ...,
10: 165297834, 1469283166, 4366441128, 4721384790, 1495900107, 59520825, 0, ...,
11: 1602117468, 18210135416, 73231116024, 117593590752, 66519597474, 8608033980, 0, ...,
12: 15792300756, 224636864830, 1183803697278, 2675326679856, 2416610807964, 672868675017, 24325703325, 0, ...,
...

David M. Jackson and Terry I. Visentin, An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces, Chapman & Hall/CRC, circa 2000. See page 227.

Joerg Arndt, Table of n, a(n) for n = 0..1325 (rows 0..50, flattened)
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).

Columns k for 0<=k<=10 are: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.
Sum of row n is A000698(n+1).
See A267180 for nonorientable analog.
Cf. A269418, A269419.
The triangle without the zeros is A269919.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4n - 2)/3 T[n-1, g] + (2n-3)(2n-2)(2n-1)/12 T[n-2, g-1] + 1/2 Sum[(2k-1)(2(n - k)-1) T[k-1, i] T[n-k-1, g-i] , {k, 1, n-1}, {i, 0, g}])/((n+1)/6);
Table[T[n, g], {n, 0, 10}, {g, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018, after Gheorghe Coserea *)

N=20;
MEM=matrix(N+1,N+1, r,c, -1);  \\ for memoization
Q(n,g)=
{
    if (n<0,  return( (g<=0) ) ); \\ not given in paper
    if (g<0,  return( 0 ) ); \\ not given in paper
    if (n<=0, return( g==0 ) );  \\ as in paper
    my( m = MEM[n+1,g+1] );
    if ( m != -1,  return(m) );  \\ memoized value
    my( t=0 );
    t += (4*n-2)/3 * Q(n-1, g);
    t += (2*n-3)*(2*n-2)*(2*n-1)/12 * Q(n-2, g-1);
    my(l, j);
    t += 1/2*
        sum(k=1, n-1, l=n-k;  \\ l+k == n, both >= 1
            sum(i=0, g, j=g-i;  \\ i+j == g, both >= 0
                (2*k-1)*(2*l-1) * Q(k-1, i) * Q(l-1, j)
            );
        );
    t *= 6/(n+1);
    MEM[n+1, g+1] = t;  \\ memoize
    return(t);
}
for (n=0, N, for (g=0, n, print1(Q(n, g),", "); );  print(); ); /* print triangle */

From Gheorghe Coserea, Mar 11 2016: (Start)
(n+1)/6 * T(n, g) = (4*n-2)/3 * T(n-1, g) + (2*n-3)*(2*n-2)*(2*n-1)/12 * T(n-2, g-1) + 1/2 * Sum_{k=1..n-1} Sum_{i=0..g} (2*k-1) * (2*(n-k)-1) * T(k-1, i) * T(n-k-1, g-i) for all n >= 1 and 0 <= g <= n/2, with the initial conditions T(0,0) = 1 and T(n,g) = 0 for g < 0 or g > n/2.
For column g, as n goes to infinity we have T(n,g) ~ t(g) * n^(5*(g-1)/2) * 12^n, where t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)) and gamma is the Gamma function.
(End)

A007137 Number of rooted maps with n edges on the projective plane.

Original entry on oeis.org

1, 10, 98, 982, 10062, 105024, 1112757, 11934910, 129307100, 1412855500, 15548498902, 172168201088, 1916619748084, 21436209373224, 240741065193282, 2713584138389838, 30687358107371442, 348061628432108352

Offset: 1

N. J. A. Sloane

E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
David M. Jackson and Terry I. Visentin, An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces, Chapman & Hall/CRC, circa 2000. See page 227.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..100
E. A. Bender, E. R. Canfield and R. W. Robinson, The enumeration of maps on the torus and the projective plane, Canad. Math. Bull., 31 (1988), 257-271; see p. 270.
Guillaume Chapuy, Maciej Dołęga, A bijection for rooted maps on general surfaces, arXiv:1501.06942 [math.CO], 2016; see corollary 4.5.
Valery A. Liskovets, A reductive technique for enumerating non-isomorphic planar maps, Discrete Math. 156 (1996), no. 1-3, 197--217. MR1405018 (97f:05087). - _N. J. A. Sloane_, Jun 03 2012

Cf. A006300.

A column of A267180.

R:=sqrt(1-12*x): seq(coeff(convert(series(((2*R+1)/3-sqrt(R*(R+2)/3))/(2*x),x,50),polynom),x,n),n=1..25); # Pab Ter, Nov 07 2005

With[{r=Sqrt[1-12x]},Rest[CoefficientList[Series[((2r+1)/3-Sqrt[r (r+2)/3])/ (2x),{x,0,20}],x]]](* Harvey P. Dale, Mar 02 2018 *)

seq(N) = {
  my(x = 'x + O('x^(N+2)), r=sqrt(1-12*x));
  Vec(((2*r+1)/3 - sqrt(r*(r+2)/3))/(2*x));
};
seq(18)
\\ test: y = 'x*Ser(seq(300),'x); 0 == 9*x^3*y^4 - 6*x^2*y^3 + 2*x*(21*x - 1)*y^2 + (10*x - 1)*y + x
\\ Gheorghe Coserea, Jul 07 2018

b(n) = sum(k=0, n\2, n!/(k!^2 * (n - 2*k)!)); \\ A002426
a(n) = 2*sum(k=0, n-1, binomial(2*n, k) * 3^k * b(n-k))/(n+1);
vector(18, n, a(n)) \\ Gheorghe Coserea, Dec 26 2018

From Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005: (Start)
G.f.: ((2*R+1)/3-sqrt(R*(R+2)/3))/(2*x) where R=sqrt(1-12*x);
a(n) ~ sqrt(3/2)*12^n/(n^(5/4)*GAMMA(3/4)). (End)
From Gheorghe Coserea, Dec 26 2018: (Start)
a(n) = (2/(n+1)) * Sum_{k=0..n-1} binomial(2*n, k) * 3^k * A002426(n-k).
G.f. y=A(x) satisfies:
0 = 9*x^3*y^4 - 6*x^2*y^3 + 2*x*(21*x - 1)*y^2 + (10*x - 1)*y + x.
0 = x*(4*x + 1)*(12*x - 1)^3*y'''' + 4*(132*x^2 + 19*x - 1)*(12*x - 1)^2*y''' + 12*(1476*x^2 + 60*x - 11)*(12*x - 1)*y'' + 72*(2016*x^2 - 117*x - 4)*y' + 648*(16*x - 1)*y.
(End)

Reference gives 20 terms
Description corrected May 15 1997, thanks to Jean-Francois Beraud
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005

A006344 Number of rooted maps with n edges on Klein bottle.

Original entry on oeis.org

4, 84, 1340, 19280, 263284, 3486224, 45247084, 579150012, 7338291224, 92272957568, 1153361204996, 14348020042512, 177803262186064, 2196338193610064, 27057921450352204, 332583930387073712

Offset: 2

Jean-Francois Beraud (beraud(AT)univ-mlv.fr)

nonn

David M. Jackson and Terry I. Visentin, An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces, Chapman & Hall/CRC, circa 2000. See page 227.

Didier Arquès and Jean-François Béraud, Énumération des cartes pointées sur la bouteille de Klein, RAIRO, vol. 31, no. 4, 1997, pp. 385-409.

A column of A267180.

A118450 Number of rooted n-edge one-vertex one-face maps on a non-orientable surface (of genus n).

Original entry on oeis.org

1, 4, 41, 488, 8229, 164892, 4016613, 112818960

Offset: 1

Valery A. Liskovets, May 04 2006

One-vertex one-face maps on orientable surfaces are counted by A035319.

E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
D. M. Jackson and T. I. Visentin, An atlas of the smaller maps in orientable and nonorientable surfaces. CRC Press, Boca Raton, 2001.

Main diagonal of A267180.

Cf. A035319.

a(7)-a(8) added by Andrew Howroyd, Jan 16 2022

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A380235 Triangle read by rows: T(n,k) is the number of n edge non-orientable genus k maps.

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A238396 Triangle T(n,k) read by rows: T(n,k) is the number of rooted genus-k maps with n edges, n>=0, 0<=k<=n.

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A007137 Number of rooted maps with n edges on the projective plane.

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A006344 Number of rooted maps with n edges on Klein bottle.

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A118450 Number of rooted n-edge one-vertex one-face maps on a non-orientable surface (of genus n).

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