A306675 -id:A306675 - cp's OEIS frontend

A306506 Number T(n,k) of permutations p of [n] having at least one index i with |p(i)-i| = k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

1, 1, 1, 4, 4, 3, 15, 19, 15, 10, 76, 99, 86, 67, 42, 455, 603, 544, 455, 358, 216, 3186, 4248, 3934, 3486, 2921, 2250, 1320, 25487, 34115, 32079, 29296, 25487, 21514, 16296, 9360, 229384, 307875, 292509, 272064, 245806, 214551, 179058, 133800, 75600

Offset: 1

Alois P. Heinz, Feb 20 2019

T(n,k) is defined for n,k>=0. The triangle contains only the terms with k=n.

			The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have absolute displacement sets {|p(i)-i|, i=1..3}: {0}, {0,1}, {0,1}, {1,2}, {1,2}, {0,2}, respectively. Number 0 occurs four times, 1 occurs four times, and 2 occurs thrice. So row n=3 is [4, 4, 3].
Triangle T(n,k) begins:
      1;
      1,     1;
      4,     4,     3;
     15,    19,    15,    10;
     76,    99,    86,    67,    42;
    455,   603,   544,   455,   358,   216;
   3186,  4248,  3934,  3486,  2921,  2250,  1320;
  25487, 34115, 32079, 29296, 25487, 21514, 16296, 9360;
  ...

Alois P. Heinz, Rows n = 1..35, flattened
Wikipedia, Permutation

Columns k=0-3 give: A002467, A306511, A306524, A324366.
T(n+2,n+1) gives A007680 (for n>=0).
T(2n,n) gives A306675.
Cf. A000142, A010050, A306461, A306512, A306535.

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

T[n_, k_] := n!-Permanent[Table[If[Abs[i-j]==k, 0, 1], {i, 1, n}, {j, 1, n} ]];
Table[T[n, k], {n, 1, 9}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, May 01 2019, from 2nd Maple program *)

T(n,k) = n! - A306512(n,k).

T(2n,n) = T(2n,0) = A002467(2n) = (2n)! - A306535(n).

A306535 Number of permutations p of [2n] having no index i with |p(i)-i| = n.

Original entry on oeis.org

1, 1, 9, 265, 14833, 1334961, 176214841, 32071101049, 7697064251745, 2355301661033953, 895014631192902121, 413496759611120779881, 228250211305338670494289, 148362637348470135821287825, 112162153835443422680893595673, 97581073836835777732377428235481

Offset: 0

Alois P. Heinz, Feb 22 2019

nonn

Also 0th term of the 2n-th forward differences of n!.

Alois P. Heinz, Table of n, a(n) for n = 0..225
Wikipedia, Permutation

Cf. A306512, A306675.

Cf. A000142, A047920, A068106.

b:= proc(n, k) b(n, k):= `if`(k=0, n!, b(n+1, k-1) -b(n, k-1)) end:
a:= n-> b(0, 2*n):
seq(a(n), n=0..23);
seq(simplify(KummerU(-2*n, -2*n, -1)), n=0..15); # Peter Luschny, May 10 2022

b[n_, k_] := b[n, k] = If[k == 0, n!, b[n + 1, k - 1] - b[n, k - 1]];
a[n_] := b[0, 2n];
a /@ Range[0, 23] (* Jean-François Alcover, Apr 02 2021, after Alois P. Heinz *)

a(n) = A306512(2n,n).
a(n) = (2n)! - A306675(n).
a(n) = KummerU(-2*n, -2*n, -1). - Peter Luschny, May 10 2022

cp's OEIS Frontend

A306506 Number T(n,k) of permutations p of [n] having at least one index i with |p(i)-i| = k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

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