login
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.
15

%I #23 May 26 2019 14:55:11

%S 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,

%T 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,

%U 4,24,576,362880,1,1,1,1,1,1,1,2,8,72,2880,3628800,1

%N 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.

%H Alois P. Heinz, <a href="/A275062/b275062.txt">Antidiagonals n = 0..140, flattened</a>

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

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

%F For k > 0, A(n, k) ~ (2*Pi*n)^((k - 1)/2) * n! / k^(n + k/2). - _Vaclav Kotesovec_, Oct 02 2018

%e A(5,0) = A(5,5) = 1: 12345.

%e A(5,1) = 5! = 120: all permutations of {1,2,3,4,5}.

%e A(5,2) = 12: 12345, 12543, 14325, 14523, 32145, 32541, 34125, 34521, 52143, 52341, 54123, 54321.

%e A(5,3) = 4: 12345, 15342, 42315, 45312.

%e A(5,4) = 2: 12345, 52341.

%e A(7,4) = 8: 1234567, 1274563, 1634527, 1674523, 5234167, 5274163, 5634127, 5674123.

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 24, 4, 2, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 120, 12, 4, 2, 1, 1, 1, 1, 1, 1, ...

%e 1, 720, 36, 8, 4, 2, 1, 1, 1, 1, 1, ...

%e 1, 5040, 144, 24, 8, 4, 2, 1, 1, 1, 1, ...

%e 1, 40320, 576, 72, 16, 8, 4, 2, 1, 1, 1, ...

%e 1, 362880, 2880, 216, 48, 16, 8, 4, 2, 1, 1, ...

%e 1, 3628800, 14400, 864, 144, 32, 16, 8, 4, 2, 1, ...

%p A:= (n, k)-> mul(floor((n+i)/k)!, i=0..k-1):

%p seq(seq(A(n, d-n), n=0..d), d=0..14);

%t A[n_, k_] := Product[Floor[(n+i)/k]!, {i, 0, k-1}];

%t Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 26 2019, from Maple *)

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

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

%Y Cf. A225816.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Jul 15 2016