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

0,9

Links

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

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

Example

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, ...

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 *)

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.

Sequence in context: A146314 A202480 A124341 * A247005 A174215 A364457

Adjacent sequences: A275059 A275060 A275061 * A275063 A275064 A275065

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 15 2016

Status

approved