A342982 -id:A342982 - cp's OEIS frontend

A342985 Triangle read by rows: T(n,k) is the number of tree-rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

1, 0, 0, 0, 2, 0, 0, 3, 3, 0, 0, 4, 36, 4, 0, 0, 5, 135, 135, 5, 0, 0, 6, 360, 1368, 360, 6, 0, 0, 7, 798, 7350, 7350, 798, 7, 0, 0, 8, 1568, 28400, 73700, 28400, 1568, 8, 0, 0, 9, 2826, 89073, 474588, 474588, 89073, 2826, 9, 0, 0, 10, 4770, 241220, 2292790, 4818092, 2292790, 241220, 4770, 10, 0

Offset: 0

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 2 - k.

For k >= 2, column k without the initial zero term is a polynomial of degree 4*(k-2)+1.

			Triangle begins:
  1;
  0, 0;
  0, 2,    0;
  0, 3,    3,     0;
  0, 4,   36,     4,     0;
  0, 5,  135,   135,     5,     0;
  0, 6,  360,  1368,   360,     6,    0;
  0, 7,  798,  7350,  7350,   798,    7, 0;
  0, 8, 1568, 28400, 73700, 28400, 1568, 8, 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 VIIIa.

Columns and diagonals 3..5 are A006428, A006429, A006430.
Row sums are A342986.
Cf. A342980, A342982, A342984, A342987.

\\ here G(n,y) is A342984 as g.f.
F(n,y)={sum(n=0, n, x^n*sum(i=0, n, my(j=n-i); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)}
G(n,y)={my(g=F(n,y)); subst(g, x, serreverse(x*g^2))}
H(n)={my(g=G(n, y)-x*(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) + x*(1+y) is the g.f. of A342984.

A342987 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 15, 5, 0, 4, 60, 84, 14, 0, 5, 175, 650, 420, 42, 0, 6, 420, 3324, 5352, 1980, 132, 0, 7, 882, 13020, 42469, 37681, 9009, 429, 0, 8, 1680, 42240, 246540, 429120, 239752, 40040, 1430, 0, 9, 2970, 118998, 1142622, 3462354, 3711027, 1421226, 175032, 4862

Offset: 0

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 2 - k.

For k >= 2, column k is a polynomial of degree 4*(k-2)+1.

			Triangle begins:
  1;
  0, 1;
  0, 2,    2;
  0, 3,   15,     5;
  0, 4,   60,    84,     14;
  0, 5,  175,   650,    420,     42;
  0, 6,  420,  3324,   5352,   1980,    132;
  0, 7,  882, 13020,  42469,  37681,   9009,   429;
  0, 8, 1680, 42240, 246540, 429120, 239752, 40040, 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 VIIIb.

Columns k=1..4 are A000007, A000027, A006470, A006471.
Diagonals are A000108, A002740, A006432, A006433.
Row sums are A342988.
Cf. A342981, A342982, A342984, A342985.

\\ here G(n,y) is A342984 as g.f.
F(n,y)={sum(n=0, n, x^n*sum(i=0, n, my(j=n-i); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)}
G(n,y)={my(g=F(n,y)); subst(g, x, serreverse(x*g^2))}
H(n)={my(g=G(n,y)-x, v=Vec(sqrt(serreverse(x/g^2)/x))); [Vecrev(t) | t<-v]}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

G.f.: A(x,y) satisfies A(x,y) = G(x*A(x,y)^2,y) where G(x,y) + x is the g.f. of A342984.

A342984 Triangle read by rows: T(n,k) is the number of nonseparable tree-rooted planar maps with n edges and k faces, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 0, 3, 3, 0, 0, 4, 20, 4, 0, 0, 5, 75, 75, 5, 0, 0, 6, 210, 604, 210, 6, 0, 0, 7, 490, 3150, 3150, 490, 7, 0, 0, 8, 1008, 12480, 27556, 12480, 1008, 8, 0, 0, 9, 1890, 40788, 170793, 170793, 40788, 1890, 9, 0, 0, 10, 3300, 115500, 829920, 1565844, 829920, 115500, 3300, 10, 0

Offset: 0

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 2 - k.
A tree-rooted planar map is a planar map with a distinguished spanning tree.
For k >= 2, column k is a polynomial of degree 4*(k-2)+1.

			Triangle begins:
  1;
  1, 1;
  0, 2,    0;
  0, 3,    3,     0;
  0, 4,   20,     4,     0;
  0, 5,   75,    75,     5,     0;
  0, 6,  210,   604,   210,     6,    0;
  0, 7,  490,  3150,  3150,   490,    7, 0;
  0, 8, 1008, 12480, 27556, 12480, 1008, 8, 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 IVa.

Columns (and diagonals) 3..5 are A006411, A006412, A006413.
Row sums are A004304.
Cf. A342982, A342985, A342987.

\\ here F(n,y) gives A342982 as g.f.
F(n,y)={sum(n=0, n, x^n*sum(i=0, n, my(j=n-i); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)}
H(n)={my(g=F(n,y), v=Vec(subst(g, x, serreverse(x*g^2)))); 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 F(x,y) = A(x*F(x,y)^2,y) where F(x,y) is the g.f. of A342982.

A342983 Number of tree-rooted planar maps with n+1 vertices and n+1 faces.

Original entry on oeis.org

1, 6, 280, 23100, 2522520, 325909584, 47117214144, 7383099180600, 1229149289511000, 214527522662653200, 38887279926227853120, 7271332144993605081120, 1395321310426879365566400, 273697641660657106322640000, 54708248601655917595233984000

Offset: 0

Andrew Howroyd, Apr 03 2021

nonn

The number of edges is 2*n.

Also, a(n) is the number of discrete walks that start and stop at the origin, never pass below the x-axis nor to the left of the y-axis, and, in any order, have n steps that increment x, n steps that decrement x, n steps that increment y, and n steps that decrement y. It is in this sense a way to generalize the 2n-step one-dimensional walks counted by A000108 to a count in two dimensions. Proof: There are A001448(n) ways to interleave two length-2n Dyck words (A000108(n)^2) - Lee A. Newberg, Nov 17 2023

Andrew Howroyd, Table of n, a(n) for n = 0..200

Central coefficients of A342982.
Even bisection of A215288.
Cf. A000108, A001246, A001448.

PARI
```
a(n) = {(4*n)!/(n!*(n+1)!)^2}
```

a(n) = (4*n)!/(n!*(n+1)!)^2.

a(n) = A000108(n)^2 * A001448(n) = A001246(n) * A001448(n). - Alois P. Heinz, Aug 02 2023

cp's OEIS Frontend

A342985 Triangle read by rows: T(n,k) is the number of tree-rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

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A342987 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

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A342984 Triangle read by rows: T(n,k) is the number of nonseparable tree-rooted planar maps with n edges and k faces, n >= 0, k = 1..n+1.

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A342983 Number of tree-rooted planar maps with n+1 vertices and n+1 faces.

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PARI

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