A343090 -id:A343090 - cp's OEIS frontend

A343092 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without isthmuses, n >= 2, k = 1..n-1.

1, 4, 10, 10, 79, 70, 20, 340, 900, 420, 35, 1071, 5846, 7885, 2310, 56, 2772, 26320, 71372, 59080, 12012, 84, 6258, 93436, 431739, 706068, 398846, 60060, 120, 12768, 280120, 2000280, 5494896, 6052840, 2499096, 291720, 165, 24090, 739420, 7643265, 32055391, 58677420, 46759630, 14805705, 1385670

Offset: 2

Andrew Howroyd, Apr 04 2021

The number of vertices is n - k.

Column k is a polynomial of degree 3*k. This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.

			Triangle begins:
   1;
   4,   10;
  10,   79,    70;
  20,  340,   900,    420;
  35, 1071,  5846,   7885,   2310;
  56, 2772, 26320,  71372,  59080,  12012;
  84, 6258, 93436, 431739, 706068, 398846, 60060;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276
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, Table VId.

Columns 1..2 are A000292, A006469.
Diagonals are A002802, A006425, A006426, A006427.
Row sums are A343093.
Cf. A269921, A342981, A342989, A343090.

\\ Needs F from A342989.
G(n,m,y,z)={my(p=F(n,m,y,z)); subst(p, x, serreverse(x*p^2))}
H(n, g=1)={my(q=G(n, g, 'y, 'z)-x, v=Vec(polcoef(sqrt(serreverse(x/q^2)/x), g, 'y))); [Vecrev(t) | t<-v]}
{ my(T=H(10)); for(n=1, #T, print(T[n])) }

A342980 Triangle read by rows: T(n,k) is the number of rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 8, 1, 0, 0, 1, 20, 20, 1, 0, 0, 1, 38, 131, 38, 1, 0, 0, 1, 63, 469, 469, 63, 1, 0, 0, 1, 96, 1262, 3008, 1262, 96, 1, 0, 0, 1, 138, 2862, 12843, 12843, 2862, 138, 1, 0, 0, 1, 190, 5780, 42602, 83088, 42602, 5780, 190, 1, 0

Offset: 0

Andrew Howroyd, Apr 01 2021

The number of vertices is n + 2 - k.

For k >= 2, columns k without the initial zero term is a polynomial of degree 3*(k-2). This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.

			Triangle begins:
  1;
  0, 0;
  0, 1,   0;
  0, 1,   1,    0;
  0, 1,   8,    1,     0;
  0, 1,  20,   20,     1,     0;
  0, 1,  38,  131,    38,     1,    0;
  0, 1,  63,  469,   469,    63,    1,   0;
  0, 1,  96, 1262,  3008,  1262,   96,   1, 0;
  0, 1, 138, 2862, 12843, 12843, 2862, 138, 1, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
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, Table VIa.

Columns (and diagonals) are A006416, A006417, A006418.
Row sums are A099553(n+1).
Cf. A082680, A269920, A342981, A342985, A343090.

G[m_, y_] := Sum[x^n*Sum[(n + k - 1)!*(2*n - k)!*y^k/(k!*(n + 1 - k)!*(2*k - 1)!*(2*n - 2*k + 1)!), {k, 1, n}], {n, 1, m}] + O[x]^m;
H[n_] := With[{g = 1 + x*G[n - 1, y]}, Sqrt[InverseSeries[x/g^2 + O[x]^(n + 1), x]/x]];
Join[{{1}, {0, 0}}, Append[CoefficientList[#, y], 0]& /@ CoefficientList[ H[11], x][[3;;]]] // Flatten (* Jean-François Alcover, Apr 15 2021, after Andrew Howroyd *)

\\ here G(n,y) gives A082680 as g.f.
G(n,y)={sum(n=1, n, x^n*sum(k=1, n, (n+k-1)!*(2*n-k)!*y^k/(k!*(n+1-k)!*(2*k-1)!*(2*n-2*k+1)!))) + O(x*x^n)}
H(n)={my(g=1+x*G(n-1, y), v=Vec(sqrt(serreverse(x/g^2)/x))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

T(n,n+2-k) = T(n,k).

G.f.: A(x,y) satisfies A(x,y) = G(x*A(x,y)^2,y) where G(x,y) = 1 + x*B(x,y) and B(x,y) is the g.f. of A082680.

A342989 Triangle read by rows: T(n,k) is the number of nonseparable rooted toroidal maps with n edges and k faces, n >= 2, k = 1..n-1.

Original entry on oeis.org

1, 4, 4, 10, 39, 10, 20, 190, 190, 20, 35, 651, 1568, 651, 35, 56, 1792, 8344, 8344, 1792, 56, 84, 4242, 33580, 64667, 33580, 4242, 84, 120, 8988, 111100, 361884, 361884, 111100, 8988, 120, 165, 17490, 317680, 1607125, 2713561, 1607125, 317680, 17490, 165

Offset: 2

Andrew Howroyd, Apr 04 2021

The number of vertices is n - k.

Column k is a polynomial of degree 3*k. This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.

			Triangle begins:
    1;
    4,    4;
   10,   39,     10;
   20,  190,    190,     20;
   35,  651,   1568,    651,     35;
   56, 1792,   8344,   8344,   1792,     56;
   84, 4242,  33580,  64667,  33580,   4242,   84;
  120, 8988, 111100, 361884, 361884, 111100, 8988, 120;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)
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, Table II.

Columns 1..4 are A000292, A006408, A006409, A006410.
Row sums are A343089.
Cf. A082680 (planar case), A269921 (rooted toroidal maps), A343090, A343092.

MQ(n,g,x=1)={ \\ after Quadric in A269921.
  my(Q=matrix(n+1,g+1)); Q[1,1]=x;
  for(n=1, n, for(g=0, min(n\2,g),
     my(t = (1+x)*(2*n-1)/3 * Q[n, 1+g]
       + if(g && n>1, (2*n-3)*(2*n-2)*(2*n-1)/12 * Q[n-1, g])
       + sum(k = 1, n-1, sum(i = 0, g, (2*k-1) * (2*(n-k)-1) * Q[k, 1+i] * Q[n-k, 1+g-i]))/2);
     Q[1+n, 1+g] = t * 6/(n+1); ));
  Q
}
F(n,m,y,z)={my(Q=MQ(n,m,z)); sum(n=0, n, x^n*Ser(Q[1+n,]/z, y)) + O(x*x^n)}
H(n,g=1)={my(p=F(n,g,'y,'z), v=Vec(polcoef(subst(p, x, serreverse(x*p^2)), g, 'y))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(10)); for(n=1, #T, print(T[n])) }

T(n,n-k) = T(n,k).

A006422 Number of rooted toroidal maps with 2 faces and n vertices and without separating cycles or isthmuses.

Original entry on oeis.org

4, 47, 240, 831, 2282, 5362, 11256, 21690, 39072, 66649, 108680, 170625, 259350, 383348, 552976, 780708, 1081404, 1472595, 1974784, 2611763, 3410946, 4403718, 5625800, 7117630, 8924760, 11098269, 13695192, 16778965, 20419886, 24695592, 29691552, 35501576

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
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.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Column 2 of A343090.

LinearRecurrence[{7,-21,35,-35,21,-7,1},{4,47,240,831,2282,5362,11256},40] (* Harvey P. Dale, May 15 2023 *)

a(n) = {n*(n + 1)*(n + 2)*(8*n^3 + 87*n^2 + 148*n - 3)/360}

From Colin Barker, Apr 09 2013: (Start)
a(n) = n*(n + 1)*(n + 2)*(8*n^3 + 87*n^2 + 148*n - 3)/360.
G.f.: x*(2*x^3+5*x^2-19*x-4) / (x-1)^7. (End)

Name clarified and terms a(11) and beyond from Andrew Howroyd, Apr 04 2021

A343091 Number of rooted toroidal maps with n edges and no separating cycles or isthmuses.

Original entry on oeis.org

1, 8, 67, 520, 3978, 29988, 224295, 1667888, 12352126, 91196512, 671717950, 4938370736, 36251576684, 265787720908, 1946696834735, 14245828801024, 104173927718694, 761302443498960, 5560564162774202, 40595140188994224, 296242813675800300, 2161031983823779912

Offset: 2

Andrew Howroyd, Apr 04 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 2..100

Row sums of A343090.

Cf. A343093.

A006423 Number of rooted toroidal maps with 3 faces and n vertices and without separating cycles or isthmuses.

Original entry on oeis.org

10, 240, 2246, 12656, 52164, 173776, 495820, 1256992, 2902702, 6214208, 12494482, 23827440, 43430088, 76120288, 128926232, 211867328, 338940050, 529346384, 809006814, 1212404336, 1784810764, 2584951600, 3688170980, 5190163680, 7211346870, 9901950240, 13447909290

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
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.

Column 3 of A343090.

a(n) = {n*(n + 1)*(n + 2)*(n + 3)*(29*n^5 + 762*n^4 + 5111*n^3 + 7902*n^2 + 8*n + 5088)/45360}

a(n) = n*(n + 1)*(n + 2)*(n + 3)*(29*n^5 + 762*n^4 + 5111*n^3 + 7902*n^2 + 8*n + 5088)/45360. - Andrew Howroyd, Apr 04 2021

Name clarified and terms a(10) and beyond from Andrew Howroyd, Apr 04 2021

A006424 Number of rooted toroidal maps with 4 faces and n vertices and without separating cycles or isthmuses.

Original entry on oeis.org

20, 831, 12656, 109075, 648792, 2978245, 11293436, 36973989, 107727724, 285451894, 699013380, 1601397330, 3465135024, 7135903782, 14072047976, 26707904230, 48991682628, 87164772761, 150869282184, 254695011933, 420306632200, 679327313795, 1077197343300, 1678276223715

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
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.

Column 4 of A343090.

a(n) = {n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(136*n^7 + 6554*n^6 + 98165*n^5 + 516860*n^4 + 763064*n^3 + 306776*n^2 + 2220955*n - 586110)/19958400} \\ Andrew Howroyd, Apr 04 2021

a(n) = n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(136*n^7 + 6554*n^6 + 98165*n^5 + 516860*n^4 + 763064*n^3 + 306776*n^2 + 2220955*n - 586110)/19958400. - Andrew Howroyd, Apr 04 2021

Name clarified and terms a(9) and beyond from Andrew Howroyd, Apr 04 2021

cp's OEIS Frontend

A343092 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without isthmuses, n >= 2, k = 1..n-1.

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A342980 Triangle read by rows: T(n,k) is the number of rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

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A342989 Triangle read by rows: T(n,k) is the number of nonseparable rooted toroidal maps with n edges and k faces, n >= 2, k = 1..n-1.

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A006422 Number of rooted toroidal maps with 2 faces and n vertices and without separating cycles or isthmuses.

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A343091 Number of rooted toroidal maps with n edges and no separating cycles or isthmuses.

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A006423 Number of rooted toroidal maps with 3 faces and n vertices and without separating cycles or isthmuses.

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A006424 Number of rooted toroidal maps with 4 faces and n vertices and without separating cycles or isthmuses.

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