A161133 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having exactly k odd fixed points (0 <= k <= ceiling(n/2)).

1, 0, 1, 1, 1, 3, 2, 1, 14, 8, 2, 64, 42, 12, 2, 426, 234, 54, 6, 2790, 1704, 468, 72, 6, 24024, 12864, 3024, 384, 24, 205056, 120120, 32160, 5040, 480, 24, 2170680, 1145400, 272400, 37200, 3000, 120, 22852200, 13024080, 3436200, 544800, 55800

Offset: 0

Author

Emeric Deutsch, Jul 18 2009

Keywords

• nonn
• tabf

Comments

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

Sum of row n is n! = A000142(n).

T(n,0) = A161131(n).

Sum_{k>=0} k*T(n,k) = A052558(n-1).

Examples

			T(3,0)=3 because we have 312, 231, 321; T(3,2)=1 because we have 123.
Triangle starts:
    1;
    0,   1;
    1,   1;
    3,   2,  1;
   14,   8,  2;
   64,  42, 12, 2;
  426, 234, 54, 6;
		

Links

• Indranil Ghosh, Rows 0..100, flattened

Crossrefs

Cf. A000142, A161131, A052558, A161134.

Programs

Maple

T := proc (n, k) options operator, arrow: binomial(ceil((1/2)*n), k)*add((-1)^j*binomial(ceil((1/2)*n)-k, j)*factorial(n-k-j), j = 0 .. ceil((1/2)*n)-k) end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do; # yields sequence in triangular form

Mathematica

Flatten[Table[Binomial[Ceiling[n/2], k]*Sum[(-1)^j*(n - k - j)!*Binomial[Ceiling[n/2] - k, j], {j, 0, Ceiling[n/2] - k}],{n, 0, 11}, {k, 0, Ceiling[n/2]}]] (* Indranil Ghosh, Mar 08 2017 *)

PARI

tabf(nn) = { for(n=0, nn, for(k = 0, ceil(n/2), print1(binomial(ceil(n/2), k) * sum(j=0, ceil(n/2) - k, (-1)^j*(n - k - j)! * binomial(ceil(n/2) - k, j)),", ");); print();); };
tabf(12); \\ Indranil Ghosh, Mar 08 2017

Formula

T(n,k) = binomial(ceiling(n/2), k)*Sum_{j=0..ceiling(n/2)-k} (-1)^j*(n-k-j)!*binomial(ceiling(n/2)-k, j).