A161134 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having exactly k even fixed points (0 <= k <= floor(n/2)).

1, 1, 1, 1, 4, 2, 14, 8, 2, 78, 36, 6, 426, 234, 54, 6, 3216, 1512, 288, 24, 24024, 12864, 3024, 384, 24, 229080, 108960, 22320, 2400, 120, 2170680, 1145400, 272400, 37200, 3000, 120, 25022880, 11998800, 2563200, 309600, 21600, 720, 287250480

Offset

0,5

Comments

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

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

T(n,0) = A161132(n).

Sum_{k>=0} k*T(n,k) = A052591(n-1).

Links

Indranil Ghosh, Rows 0..125, flattened

Formula

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

Example

T(3,0)=4 because we have 132, 312, 213, 231; T(3,1)=2 because we have 123 and 321.
Triangle starts:
    1;
    1;
    1,   1;
    4,   2;
   14,   8,   2;
   78,  36,   6;
  426, 234,  54,   6;

Maple

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

Mathematica

Flatten[Table[Binomial[Floor[n/2], k]*Sum[(-1)^j*(n - k - j)!*Binomial[Floor[n/2] - k, j], {j, 0, Floor[n/2] - k}], {n, 0, 12}, {k, 0, Floor[n/2]}]] (* Indranil Ghosh, Mar 08 2017 *)

Prog

(PARI) tabf(nn) = { for(n=0, nn, for(k = 0, floor(n/2), print1(binomial(floor(n/2), k) * sum(j=0, floor(n/2) - k, (-1)^j*(n - k - j)! * binomial(floor(n/2) - k, j)), ", "); ); print(); ); };
tabf(12); \\ Indranil Ghosh, Mar 08 2017
(Python)
from sympy import factorial, binomial
def T(n, k):
    s=0
    for j in range(n//2 - k+1):
        s+=(-1)**j * factorial(n-k-j) * binomial(n//2 - k, j)
    return binomial(n//2, k)* s
i=0
for n in range(26):
    for k in range(n//2 + 1):
        print(str(i)+" "+str(T(n, k)))
        i+=1
# Indranil Ghosh, Mar 08 2017

Crossrefs

Cf. A000142, A052591, A161132, A161133.

Sequence in context: A330314 A255044 A248249 * A184183 A317738 A224893

Adjacent sequences: A161131 A161132 A161133 * A161135 A161136 A161137

Keyword

nonn,tabf

Author

Emeric Deutsch, Jul 18 2009

Status

approved