A275062 Number A(n,k) of permutations p of [n] such that p(i)-i is a multiple of k for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 24, 1, 1, 1, 1, 1, 4, 120, 1, 1, 1, 1, 1, 2, 12, 720, 1, 1, 1, 1, 1, 1, 4, 36, 5040, 1, 1, 1, 1, 1, 1, 2, 8, 144, 40320, 1, 1, 1, 1, 1, 1, 1, 4, 24, 576, 362880, 1, 1, 1, 1, 1, 1, 1, 2, 8, 72, 2880, 3628800, 1

Offset: 0

Author

Alois P. Heinz, Jul 15 2016

Keywords

• nonn
• tabl

Examples

			A(5,0) = A(5,5) = 1: 12345.
A(5,1) = 5! = 120: all permutations of {1,2,3,4,5}.
A(5,2) = 12: 12345, 12543, 14325, 14523, 32145, 32541, 34125, 34521, 52143, 52341, 54123, 54321.
A(5,3) = 4: 12345, 15342, 42315, 45312.
A(5,4) = 2: 12345, 52341.
A(7,4) = 8: 1234567, 1274563, 1634527, 1674523, 5234167, 5274163, 5634127, 5674123.
Square array A(n,k) begins:
  1,       1,     1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,       1,     1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,       2,     1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,       6,     2,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,      24,     4,   2,   1,  1,  1, 1, 1, 1, 1, ...
  1,     120,    12,   4,   2,  1,  1, 1, 1, 1, 1, ...
  1,     720,    36,   8,   4,  2,  1, 1, 1, 1, 1, ...
  1,    5040,   144,  24,   8,  4,  2, 1, 1, 1, 1, ...
  1,   40320,   576,  72,  16,  8,  4, 2, 1, 1, 1, ...
  1,  362880,  2880, 216,  48, 16,  8, 4, 2, 1, 1, ...
  1, 3628800, 14400, 864, 144, 32, 16, 8, 4, 2, 1, ...
		

Links

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

Crossrefs

Columns k=0-10 give: A000012, A000142, A010551, A264557, A264635, A264656, A264701, A264791, A275063, A275064, A275065.

A(k*n,n) for k=1..4 give: A000012, A000079, A000400, A009968.

Cf. A225816.

Programs

Maple

A:= (n, k)-> mul(floor((n+i)/k)!, i=0..k-1):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

A[n_, k_] := Product[Floor[(n+i)/k]!, {i, 0, k-1}];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 26 2019, from Maple *)

Formula

A(n,k) = Product_{i=0..k-1} floor((n+i)/k)!.

A(k*n,k) = (n!)^k = A225816(k,n).

For k > 0, A(n, k) ~ (2*Pi*n)^((k - 1)/2) * n! / k^(n + k/2). - Vaclav Kotesovec, Oct 02 2018