A344901 Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.

1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 6, 8, 0, 0, 10, 24, 30, 40, 0, 0, 26, 160, 144, 180, 160, 0, 0, 76, 1140, 1120, 1008, 840, 700, 0, 0, 232, 8988, 9120, 8960, 5376, 4200, 2912, 0, 0, 764, 80864, 80892, 82080, 53760, 30240, 19656, 12768, 0, 0, 2620, 809856, 808640, 808920, 547200, 336000, 157248, 95760, 55680, 0, 0, 9496

Offset: 0

Author

Mikhail Kurkov, Jun 01 2021

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
     1;
     0,    1;
     0,    0,    2;
     2,    0,    0,    4;
     6,    8,    0,    0,   10;
    24,   30,   40,    0,    0,   26;
   160,  144,  180,  160,    0,    0, 76;
  1140, 1120, 1008,  840,  700,    0,  0, 232;
  8988, 9120, 8960, 5376, 4200, 2912,  0,   0, 764;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-1 give: A038205, A221145.

Row sums give A000142.

Main diagonal gives A000085.

Cf. A000023, A007318, A052571, A052849, A098558, A137482.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*
      binomial(n-1, j-1)*(j-1)!, j=`if`(t=1, 1..min(2, n), 3..n)))
    end:
T:= (n, k)-> binomial(n, k)*b(k, 1)*b(n-k, 0):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Oct 28 2024

Mathematica

b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[b[n-j, t]* Binomial[n-1, j-1]*(j-1)!, {j, If[t == 1, Range @ Min[2, n], Range[3, n]]}]];
T[n_, k_] := Binomial[n, k]*b[k, 1]*b[n-k, 0];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 24 2025, after Alois P. Heinz *)

Formula

T(n,k) = binomial(n,k)*A000085(k)*A038205(n-k).

From Alois P. Heinz, Oct 28 2024: (Start)

Sum_{k=0..n} k * T(n,k) = A052849(n) = A098558(n) for n>=2.

Sum_{k=0..n} (n-k) * T(n,k) = A052571(n).

Sum_{k=0..n} (-1)^k * T(n,k) = A000023(n).

T(n,0) + T(n,1) = A137482(n). (End)