A306512 Number A(n,k) of permutations p of [n] having no index i with |p(i)-i| = k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 9, 1, 1, 2, 3, 5, 44, 1, 1, 2, 6, 9, 21, 265, 1, 1, 2, 6, 14, 34, 117, 1854, 1, 1, 2, 6, 24, 53, 176, 792, 14833, 1, 1, 2, 6, 24, 78, 265, 1106, 6205, 133496, 1, 1, 2, 6, 24, 120, 362, 1554, 8241, 55005, 1334961

Offset

0,10

Links

Alois P. Heinz, Antidiagonals n = 0..45, flattened

Wikipedia, Permutation

Formula

A(n,k) = n! - A306506(n,k).

A(n,n+i) = n! for i >= 0.

Example

A(4,0) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
A(4,1) = 5: 1234, 1432, 3214, 3412, 4231.
A(4,2) = 9: 1234, 1243, 1324, 2134, 2143, 2341, 4123, 4231, 4321.
Square array A(n,k) begins:
     1,   1,    1,    1,    1,    1,    1,    1, ...
     0,   1,    1,    1,    1,    1,    1,    1, ...
     1,   1,    2,    2,    2,    2,    2,    2, ...
     2,   2,    3,    6,    6,    6,    6,    6, ...
     9,   5,    9,   14,   24,   24,   24,   24, ...
    44,  21,   34,   53,   78,  120,  120,  120, ...
   265, 117,  176,  265,  362,  504,  720,  720, ...
  1854, 792, 1106, 1554, 2119, 2790, 3720, 5040, ...

Maple

A:= proc(n, k) option remember; `if`(k>=n, n!, LinearAlgebra[
      Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)=k, 0, 1))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
b:= proc(s, k) option remember; (n-> `if`(n=0, 1, add(
      `if`(abs(i-n)=k, 0, b(s minus {i}, k)), i=s)))(nops(s))
    end:
A:= (n, k)-> `if`(k>=n, n!, b({$1..n}, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

A[n_, k_] := If[k > n, n!, Permanent[Table[If[Abs[i-j] == k, 0, 1], {i, 1, n}, {j, 1, n}]]]; A[0, 0] = 1;
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 05 2021, from first Maple program *)
b[s_, k_] := b[s, k] = With[{n = Length[s]}, If[n == 0, 1, Sum[
     If[Abs[i-n] == k, 0, b[s ~Complement~ {i}, k]], {i, s}]]];
A[n_, k_] := If[k >= n, n!, b[Range@n, k]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Sep 01 2021, from second Maple program *)

Crossrefs

Columns k=0-3 give: A000166, A078480, A306523, A324365.

A(n+2j,n+j) (j=0..5) give: A000142, A001564, A001688, A023043, A023045, A023047.

A(2n,n) gives A306535.

Cf. A306506.

Sequence in context: A114551 A191620 A214751 * A343938 A239397 A307884

Adjacent sequences: A306509 A306510 A306511 * A306513 A306514 A306515

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 20 2019

Status

approved