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

Author

Andrew Howroyd, Apr 02 2021

Keywords

• nonn
• tabl

Comments

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.

Examples

			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;
  ...
		

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 VIb.

Crossrefs

Columns k=3..4 are A005581, A006468.

Diagonals are A000108, A006419, A006420, A006421.

Row sums are A000260.

Cf. A002293, A027836, A082680, A269920, A342980, A343092.

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*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 *)

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*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])) }

Formula

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).