A306234 Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n] divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 5, 13, 15, 13, 5, 1, 1, 7, 28, 67, 76, 67, 28, 7, 1, 1, 9, 49, 179, 411, 455, 411, 179, 49, 9, 1, 1, 11, 76, 375, 1306, 2921, 3186, 2921, 1306, 375, 76, 11, 1, 1, 13, 109, 679, 3181, 10757, 23633, 25487, 23633, 10757, 3181, 679, 109, 13, 1

Offset

1,6

Links

Alois P. Heinz, Rows n = 1..142, flattened

Wikipedia, Permutation

Formula

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

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

T(n,k) = (n-|k|)! [x^(n-|k|)] (1-exp(-x))/(1-x)^(|k|+1).

T(n+1,n) = 1.

T(n,k) = A306461(n,k) / |k|!.

Sum_{k=1-n..n-1} |k|! * T(n,k) = A306455(n).

Example

Triangle T(n,k) begins:
  :                                 1                              ;
  :                           1,    1,    1                        ;
  :                     1,    3,    4,    3,    1                  ;
  :               1,    5,   13,   15,   13,    5,   1             ;
  :          1,   7,   28,   67,   76,   67,   28,   7,  1         ;
  :      1,  9,  49,  179,  411,  455,  411,  179,  49,  9,  1     ;
  :  1, 11, 76, 375, 1306, 2921, 3186, 2921, 1306, 375, 76, 11, 1  ;

Maple

b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),
      add(b(s minus {i}, d union {n-i}), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, {})):
seq(T(n), n=1..8);
# second Maple program:
T:= (n, k)-> -add((-1)^j*binomial(n-abs(k), j)*(n-j)!, j=1..n)/abs(k)!:
seq(seq(T(n, k), k=1-n..n-1), n=1..9);

Mathematica

T[n_, k_] := (-1/Abs[k]!) Sum[(-1)^j Binomial[n-Abs[k], j] (n-j)!, {j, 1, n}];
Table[T[n, k], {n, 1, 9}, {k, 1-n, n-1}] // Flatten (* Jean-François Alcover, Feb 15 2021 *)

Crossrefs

Columns k=0-10 give (offsets may differ): A002467, A180191, A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360.

Row sums give A306525.

T(n+1,n) gives A000012.

T(n+2,n) gives A005408.

T(n+2,n-1) gives A056107.

T(2n,n) gives A324361.

Cf. A000142, A306455, A306461, A324224, A324362.

Sequence in context: A111028 A201162 A096646 * A290057 A249790 A302713

Adjacent sequences: A306231 A306232 A306233 * A306235 A306236 A306237

Keyword

nonn,tabf

Author

Alois P. Heinz, Feb 17 2019

Status

approved