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.

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

Author

Andrew Howroyd, Apr 03 2021

Keywords

• nonn
• tabl

Comments

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.

Examples

			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;
  ...
		

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 IVa.

Crossrefs

Columns (and diagonals) 3..5 are A006411, A006412, A006413.

Row sums are A004304.

Cf. A342982, A342985, A342987.

Programs

PARI

\\ 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])) }

Formula

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.