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.

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

Author

Andrew Howroyd, Apr 01 2021

Keywords

• nonn
• tabl

Comments

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.

Examples

			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;
  ...
		

Links

• 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.

Crossrefs

Columns (and diagonals) are A006416, A006417, A006418.

Row sums are A099553(n+1).

Cf. A082680, A269920, A342981, A342985, A343090.

Programs

Mathematica

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 *)

PARI

\\ 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])) }

Formula

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.