A343092 -id:A343092 - cp's OEIS frontend

A342981 Triangle read by rows: T(n,k) is the number of rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

1, 0, 1, 0, 1, 2, 0, 1, 7, 5, 0, 1, 16, 37, 14, 0, 1, 30, 150, 176, 42, 0, 1, 50, 449, 1104, 794, 132, 0, 1, 77, 1113, 4795, 7077, 3473, 429, 0, 1, 112, 2422, 16456, 41850, 41504, 14893, 1430, 0, 1, 156, 4788, 47832, 189183, 319320, 228810, 63004, 4862

Offset: 0

Andrew Howroyd, Apr 02 2021

The number of vertices is n + 2 - k.
For k >= 2, column k 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.
By duality, also the number of loopless rooted planar maps with n edges and k vertices.

			Triangle begins:
  1;
  0, 1;
  0, 1,   2;
  0, 1,   7,    5;
  0, 1,  16,   37,    14;
  0, 1,  30,  150,   176,    42;
  0, 1,  50,  449,  1104,   794,   132;
  0, 1,  77, 1113,  4795,  7077,  3473,   429;
  0, 1, 112, 2422, 16456, 41850, 41504, 14893, 1430;
  ...

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 VIb.

Columns k=3..4 are A005581, A006468.
Diagonals are A000108, A006419, A006420, A006421.
Row sums are A000260.
Cf. A002293, A027836, A082680, A269920, A342980, A343092.

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*y + x*G[n - 1, y]}, Sqrt[InverseSeries[x/g^2 + O[x]^(n + 1), x]/x]];
CoefficientList[#, y]& /@ CoefficientList[H[10], x] // 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*y+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])) }

G.f. A(x,y) satisfies A(x) = G(x*A(x,y)^2, y) where G(x,y) = 1 + x*y + x*B(x,y) and B(x,y) is the g.f. of A082680.
A027836(n+1) = Sum_{k=1..n+1} k*T(n,k).
A002293(n) = Sum_{k=1..n+1} k*T(n,n+2-k).

A343090 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without separating cycles or isthmuses, n >= 2, k = 1..n-1.

Original entry on oeis.org

1, 4, 4, 10, 47, 10, 20, 240, 240, 20, 35, 831, 2246, 831, 35, 56, 2282, 12656, 12656, 2282, 56, 84, 5362, 52164, 109075, 52164, 5362, 84, 120, 11256, 173776, 648792, 648792, 173776, 11256, 120, 165, 21690, 495820, 2978245, 5360286, 2978245, 495820, 21690, 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,    47,     10;
   20,   240,    240,     20;
   35,   831,   2246,    831,     35;
   56,  2282,  12656,  12656,   2282,     56;
   84,  5362,  52164, 109075,  52164,   5362,    84;
  120, 11256, 173776, 648792, 648792, 173776, 11256, 120;
  ...

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 VIc.

Columns 1..4 are A000292, A006422, A006423, A006424.
Row sums are A343091.
Cf. A269921, A342980, A342989, A343092.

\\ 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*(1+'z), 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])) }

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

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).

A343093 Number of rooted toroidal maps with n edges and no isthmuses.

Original entry on oeis.org

1, 14, 159, 1680, 17147, 171612, 1696491, 16631840, 162090756, 1572801142, 15210259585, 146710561296, 1412132981778, 13569013500024, 130199055578307, 1247825314752768, 11947157409479180, 114288613130155608, 1092495810452593564, 10436544808441964352

Offset: 2

Andrew Howroyd, Apr 04 2021

nonn

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

Row sums of A343092.

Cf. A224274, A343091.

Conjectural g.f.: A(x) = ( Sum_{n >= 1} (1/4)*binomial(4*n, n)*x^n )^2. - Peter Bala, Jul 24 2025

A006469 Number of rooted toroidal maps with 2 faces, n vertices and no isthmuses.

Original entry on oeis.org

10, 79, 340, 1071, 2772, 6258, 12768, 24090, 42702, 71929, 116116, 180817, 273000, 401268, 576096, 810084, 1118226, 1518195, 2030644, 2679523, 3492412, 4500870, 5740800, 7252830, 9082710, 11281725, 13907124, 17022565, 20698576, 25013032, 30051648, 35908488

Offset: 1

N. J. A. Sloane

A map on a torus has genus 1.

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

Colin Barker, 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 A343092.

A006469[n_] := n*(n + 1)*(n + 2)*(n + 3)*(n*(8*n + 63) + 79)/360;
Array[A006469, 50] (* Paolo Xausa, Aug 20 2025 *)

Vec(x*(10 + 9*x - 3*x^2) / (1 - x)^7 + O(x^40)) \\ Colin Barker, Apr 22 2017

G.f.: x/(x-1)^7*(3*x^2-9*x-10). - Simon Plouffe, Master's thesis, Uqam 1992
From Colin Barker, Apr 22 2017: (Start)
a(n) = (n*(474 + 1247*n + 1215*n^2 + 545*n^3 + 111*n^4 + 8*n^5)) / 360.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

Name improved by Sean A. Irvine, Apr 21 2017

A006425 Number of rooted toroidal maps with 2 vertices and n faces and no isthmuses.

Original entry on oeis.org

4, 79, 900, 7885, 59080, 398846, 2499096, 14805705, 83969600, 459868530, 2447439384, 12718070274, 64766697520, 324156347260, 1598200903280, 7776728909121, 37404399901296, 178060831286890, 839857764202520, 3928581810398630, 18239060530882224, 84101317494787684

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).

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.

A diagonal of A343092.

Name clarified and terms a(11) and beyond from Andrew Howroyd, Apr 05 2021

A006426 Number of rooted toroidal maps with 3 vertices and n faces and no isthmuses.

Original entry on oeis.org

10, 340, 5846, 71372, 706068, 6052840, 46759630, 333746556, 2238411692, 14277544216, 87376309020, 516495616120, 2964332933800, 16586670357200, 90782049175614, 487329793111260, 2571575908919740, 13364166071956280, 68507393061864020, 346874109053120616

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).

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.

A diagonal of A343092.

Name clarified and terms a(10) and beyond from Andrew Howroyd, Apr 05 2021

A006427 Number of rooted toroidal maps with 4 vertices and n faces and no isthmuses.

Original entry on oeis.org

20, 1071, 26320, 431739, 5494896, 58677420, 550712668, 4681144391, 36786186216, 271221867098, 1896796135920, 12688048319278, 81709791432384, 509222462694582, 3083998029716868, 18213177504318335, 105186858991413976, 595499805083872458, 3311524095424508480

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).

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.

A diagonal of A343092.

Name clarified and terms a(9) and beyond from Andrew Howroyd, Apr 05 2021

cp's OEIS Frontend

A342981 Triangle read by rows: T(n,k) is the number of rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

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A343090 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without separating cycles or isthmuses, n >= 2, 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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A343093 Number of rooted toroidal maps with n edges and no isthmuses.

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A006469 Number of rooted toroidal maps with 2 faces, n vertices and no isthmuses.

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A006425 Number of rooted toroidal maps with 2 vertices and n faces and no isthmuses.

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A006426 Number of rooted toroidal maps with 3 vertices and n faces and no isthmuses.

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A006427 Number of rooted toroidal maps with 4 vertices and n faces and no isthmuses.

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