A342982 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges and k+1 faces, n >= 0, k = 0..n.

1, 1, 1, 2, 6, 2, 5, 30, 30, 5, 14, 140, 280, 140, 14, 42, 630, 2100, 2100, 630, 42, 132, 2772, 13860, 23100, 13860, 2772, 132, 429, 12012, 84084, 210210, 210210, 84084, 12012, 429, 1430, 51480, 480480, 1681680, 2522520, 1681680, 480480, 51480, 1430

Offset: 0

Author

Andrew Howroyd, Apr 03 2021

Keywords

• nonn
• tabl

Comments

The number of vertices is n + 1 - k.

A tree-rooted planar map is a planar map with a distinguished spanning tree.

Examples

			Triangle begins:
    1;
    1,     1;
    2,     6,     2;
    5,    30,    30,      5;
   14,   140,   280,    140,     14;
   42,   630,  2100,   2100,    630,    42;
  132,  2772, 13860,  23100,  13860,  2772,   132;
  429, 12012, 84084, 210210, 210210, 84084, 12012, 429;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
• R. C. Mullin, On the Enumeration of Tree-Rooted Maps, Canadian Journal of Mathematics, Volume 19, 1967, pp. 174-183.

Crossrefs

Columns k=0..2 are A000108, A002457, 2*A002803.

Row sums are A005568.

Central coefficients are A342983.

Cf. A001263, A215288, A269920, A342984.

Programs

Mathematica

Table[(2 n)!/(k!*(k + 1)!*(n - k)!*(n - k + 1)!), {n, 0, 8}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2021 *)

PARI

T(n,k) = {(2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!)}
{ for(n=0, 10, print(vector(n+1, k, T(n,k-1)))) }

Formula

T(n,k) = (2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!).

T(n,n-k) = T(n,k).

T(n, floor(n/2)) = A215288(n).

T(n,k) = A000108(n) * A001263(n+1,k+1). - Werner Schulte, Apr 04 2021