A382776 Triangle read by rows: T(n,k) is the number of ways to place 2*n rooks on a (n+k) X (2*n-k) board so that there is at least one rook in every column and row and so that each rook is defended by another.

1, 1, 1, 6, 9, 6, 90, 180, 180, 90, 2520, 6300, 8100, 6300, 2520, 113400, 340200, 529200, 529200, 340200, 113400, 7484400, 26195400, 47628000, 57153600, 47628000, 26195400, 7484400, 681080400, 2724321600, 5658206400, 7858620000, 7858620000, 5658206400, 2724321600, 681080400

Offset: 0

Author

Andrew Howroyd, Apr 04 2025

Keywords

• nonn
• tabl

Comments

The configurations are such that k columns will each contain 2 rooks and n-k rows will each contain 2 rooks.

Examples

			Triangle begins:
        1;
        1,        1;
        6,        9,        6;
       90,      180,      180,       90;
     2520,     6300,     8100,     6300,     2520;
   113400,   340200,   529200,   529200,   340200,   113400;
  7484400, 26195400, 47628000, 57153600, 47628000, 26195400, 7484400;
  ...
The T(2,0) = 6 configurations are:
  X X . .    X . X .    X . . X    . X X .    . X . X    . . X X
  . . X X    . X . X    . X X .    X . . X    X . X .    X X . .
The T(2,1) = 9 configurations are:
  X X .   X . X   . X X   . . X   . X .   X . .   . . X   . X .   X . .
  . . X   . X .   X . .   X X .   X . X   . X X   . . X   . X .   X . .
  . . X   . X .   X . .   . . X   . X .   X . .   X X .   X . X   . X X
		

Crossrefs

Row sums are A382777.

Column k=0 is A000680.

Programs

PARI

T(n,k)=binomial(2*n-k,k)*binomial(n+k,n-k)*(2*(n-k))!*(2*k)!/(2^n)

Formula

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

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