A213279 Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of permutations of [1..n] in which all cycle lengths are divisible by k.

1, 2, 1, 6, 0, 2, 24, 9, 0, 6, 120, 0, 0, 0, 24, 720, 225, 160, 0, 0, 120, 5040, 0, 0, 0, 0, 0, 720, 40320, 11025, 0, 6300, 0, 0, 0, 5040, 362880, 0, 62720, 0, 0, 0, 0, 0, 40320, 3628800, 893025, 0, 0, 435456, 0, 0, 0, 0, 362880, 39916800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..141, flattened

E. D. Bolker and A. M. Gleason, Counting permutations, J. Combin. Thy., A 29 (1980), 236-242.

Example

Triangle begins
[1],
[2, 1],
[6, 0, 2],
[24, 9, 0, 6],
[120, 0, 0, 0, 24],
[720, 225, 160, 0, 0, 120],
[5040, 0, 0, 0, 0, 0, 720],
[40320, 11025, 0, 6300, 0, 0, 0, 5040],
[362880, 0, 62720, 0, 0, 0, 0, 0, 40320],
[3628800, 893025, 0, 0, 435456, 0, 0, 0, 0, 362880],
[39916800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800],
[479001600, 108056025, 68992000, 56133000, 0, 46569600, 0, 0, 0, 0, 0, 39916800],
...

Maple

read transforms;
f:=(n, d)->mul(n-j+did(j, d), j=1..n); # did(d, j) = 1 iff j divides d, otherwise 0
g:=n->[seq(f(n, d), d=1..n)];
[seq(g(n), n=1..14)];
# second Maple program:
T:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(irem(j, k)=0, binomial(n-1, j-1)*(j-1)!*
       T(n-j, k), 0), j=1..n))
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, May 14 2016

Mathematica

T[n_, k_] := T[n, k] = If[n == 0, 1, Sum[If[Mod[j, k] == 0, Binomial[n-1, j -1] * (j-1)! * T[n-j, k], 0], {j, 1, n}]];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 24 2016, after Alois P. Heinz *)

Crossrefs

Cf. A213280.

Sequence in context: A215219 A335931 A091615 * A175591 A187784 A370901

Adjacent sequences: A213276 A213277 A213278 * A213280 A213281 A213282

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 08 2012

Status

approved