A306523 -id:A306523 - cp's OEIS frontend

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

Alois P. Heinz, Feb 20 2019

			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, ...

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

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.

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);

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 *)

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

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

A306524 Number of permutations p of [n] having at least one index i with |p(i)-i| = 2.

Original entry on oeis.org

0, 0, 0, 3, 15, 86, 544, 3934, 32079, 292509, 2952702, 32712087, 394749367, 5155010088, 72440184064, 1090017765544, 17486996858151, 297965879586295, 5374189975316350, 102290550351854445, 2049025241258716927, 43089888746430771294, 949172134240270646352

Offset: 0

Alois P. Heinz, Feb 21 2019

nonn

			a(3) = 3: 231, 312, 321.
a(4) = 15: 1342, 1423, 1432, 2314, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4312.

Alois P. Heinz, Table of n, a(n) for n = 0..450
Wikipedia, Permutation

Column k=2 of A306506.

Cf. A000142, A306523.

T[n_, k_] := n! - Permanent[Table[If[Abs[i-j] == k, 0, 1], {i, 1, n}, {j, 1, n}]];
a[n_] := If[n == 0, 0, T[n, 2]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 31 2021, after Alois P. Heinz in A306506 *)

a(n) = n! - A306523(n).

cp's OEIS Frontend

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.

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