A152662 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} for which k is the maximal number of initial odd entries (0 <= k <= ceiling(n/2)).

1, 0, 1, 1, 1, 2, 2, 2, 12, 8, 4, 48, 36, 24, 12, 360, 216, 108, 36, 2160, 1440, 864, 432, 144, 20160, 11520, 5760, 2304, 576, 161280, 100800, 57600, 28800, 11520, 2880, 1814400, 1008000, 504000, 216000, 72000, 14400, 18144000, 10886400, 6048000, 3024000

Offset: 0

Author

Emeric Deutsch, Dec 13 2008

Keywords

• nonn
• tabf

Comments

Sum of entries in row n is n! (A000142).

Row n has 1 + ceiling(n/2) entries.

T(n,0) = A052591(n-1) for n>=1.

Sum_{k=0..ceiling(n/2)} k*T(n,k) = A152663(n).

Examples

			T(3,0)=2 because we have 213 and 231.
T(4,2)=4 because we have 1324, 1342, 3124 and 3142.
Triangle starts:
    1;
    0,   1;
    1,   1;
    2,   2,   2;
   12,   8,   4;
   48,  36,  24,  12;
  360, 216, 108,  36;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Cf. A000142, A010551, A052591, A152663, A152664, A256881.

Programs

Maple

T := proc (n, k) if n=0 and k=0 then 1 elif n = 1 and k = 0 then 0 elif n = 1 and k = 1 then 1 elif `mod`(n, 2) = 1 then (1/2)*(n-1)*binomial((1/2)*n+1/2, k)*factorial(k)*factorial(n-1-k) else (1/2)*n*binomial((1/2)*n, k)*factorial(k)*factorial(n-1-k) end if end proc: for n from 0 to 11 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do; # yields sequence in triangular form

Mathematica

T[n_, k_] := T[n, k] = Which[n == 0 && k == 0, 1, n == 1 && k == 1, 1, OddQ[n], (n - 1)/2*k!*(n - k - 1)!*Binomial[(n - 1)/2 + 1, k], True, n/2*k!*(n - k - 1)!*Binomial[n/2, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, Ceiling[n/2]}] // Flatten (* Jean-François Alcover, Apr 04 2024 *)

Formula

T(2n+1,k) = n*k!*(2n-k)!*binomial(n+1,k) (n>= 1);

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

From Alois P. Heinz, Apr 02 2024: (Start)

Sum_{k>=0} (k+1) * T(n,k) = A256881(n+1).

T(n,ceiling(n/2)) = A010551(n). (End)

Extensions

T(0,0)=1 prepended by Alois P. Heinz, Apr 02 2024