A269921 -id:A269921 - cp's OEIS frontend

A288072 a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 1.

2310, 100156, 2278660, 36703824, 472592916, 5188948072, 50534154408, 448035881592, 3682811916980, 28442316247080, 208462422428152, 1461307573813824, 9857665477085832, 64309102366765200, 407372683115470800, 2514120288996270024, 15159074541052024308, 89512241718624419624

Offset: 6

Gheorghe Coserea, Jun 05 2017

nonn

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

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, this sequence, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
Column 5 of A269921.
Cf. A000108.

Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n + 1) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 Q[n - 2, f, g - 1] + 1/2 Sum[l = n - k; Sum[v = f - u; Sum[j = g - i; Boole[l >= 1 && v >= 1 && j >= 0] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 5, 1];
Table[a[n], {n, 6, 23}] (* Jean-François Alcover, Oct 18 2018 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288072_ser(N) = {
  my(y = A000108_ser(N+1));
  -2*y*(y-1)^6*(2140*y^5 + 14751*y^4 - 15604*y^3 - 8820*y^2 + 10176*y - 1488)/(y-2)^17;
};
Vec(A288072_ser(18))

A288073 a(n) is the number of rooted maps with n edges and 9 faces on an orientable surface of genus 1.

Original entry on oeis.org

1385670, 126264820, 5593305476, 164767964504, 3682811916980, 67173739068760, 1046677747672360, 14373136466094880, 177882700353757460, 2017523504473479992, 21241931655650633720, 209732362862241103248, 1957830216739337392584, 17394726697224718134384, 147908195064869691109072

Offset: 10

Gheorghe Coserea, Jun 05 2017

nonn

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

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, this sequence, A288074 f=10.

Column 9 of A269921.

Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n + 1) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 Q[n - 2, f, g - 1] + 1/2 Sum[l = n - k; Sum[v = f - u; Sum[j = g - i; Boole[l >= 1 && v >= 1 && j >= 0] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 9, 1];
Table[a[n], {n, 10, 24}] (* Jean-François Alcover, Oct 18 2018 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288073_ser(N) = {
  my(y = A000108_ser(N+1));
  -2*y*(y-1)^10*(58911256*y^9 + 315266323*y^8 - 563073084*y^7 - 706445836*y^6 + 1588166368*y^5 - 488205920*y^4 - 472512192*y^3 + 315108288*y^2 - 44342784*y - 2179584)/(y-2)^29;
};
Vec(A288073_ser(17))

A288074 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 1.

Original entry on oeis.org

6466460, 678405090, 34225196720, 1137369687454, 28442316247080, 576218752277476, 9908748651241088, 149314477245194262, 2017523504473479992, 24868664942648145372, 283389619978690157408, 3017066587822315930220, 30265092793614787511376, 288055728071446557904968, 2616366012933033221518720

Offset: 11

Gheorghe Coserea, Jun 05 2017

nonn

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

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, this sequence.

Column 10 of A269921.

Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n + 1) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 Q[n - 2, f, g - 1] + 1/2 Sum[l = n - k; Sum[v = f - u; Sum[j = g - i; Boole[l >= 1 && v >= 1 && j >= 0] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 10, 1];
Table[a[n], {n, 11, 25}] (* Jean-François Alcover, Oct 18 2018 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288074_ser(N) = {
  my(y = A000108_ser(N+1));
  2*y*(y-1)^11*(734641583*y^10 + 3795452665*y^9 - 7483071778*y^8 - 10235465624*y^7 + 25178445968*y^6 - 7563355856*y^5 - 11624244832*y^4 + 8854962048*y^3 - 1433163264*y^2 - 286758144*y + 65790464)/(y-2)^32;
};
Vec(A288074_ser(15))

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.

Original entry on oeis.org

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])) }

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).

A006297 Number of rooted genus-1 maps with n edges.

Original entry on oeis.org

1, 10, 167, 1720, 24164, 256116, 3392843, 36703824, 472592916, 5188948072, 65723863196, 729734918432, 9145847808784, 102432266545800, 1274461449989715, 14373136466094880, 177882700353757460, 2017523504473479992, 24868664942648145372

Offset: 2

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, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

Row maxima of A269921.

More terms from Sean A. Irvine, Feb 24 2017

cp's OEIS Frontend

A288072 a(n) is the number of rooted maps with n edges and 5 faces on an orientable surface of genus 1.

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A288073 a(n) is the number of rooted maps with n edges and 9 faces on an orientable surface of genus 1.

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A288074 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 1.

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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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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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A006297 Number of rooted genus-1 maps with n edges.

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