A335109 Triangle read by rows: T(n,k) is the number of permutations of length n with each cycle of the permutation containing only elements that are identical (mod k), where 1 <= k <= n.

1, 2, 1, 6, 2, 1, 24, 4, 2, 1, 120, 12, 4, 2, 1, 720, 36, 8, 4, 2, 1, 5040, 144, 24, 8, 4, 2, 1, 40320, 576, 72, 16, 8, 4, 2, 1, 362880, 2880, 216, 48, 16, 8, 4, 2, 1, 3628800, 14400, 864, 144, 32, 16, 8, 4, 2, 1

Offset: 1

Author

Dennis P. Walsh, May 23 2020

Keywords

• nonn
• tabl

Comments

Let [n] denote {1,2,...,n} and let [n](j,k) denote the subset of [n] consisting of all elements of [n] that equal j mod k. The cardinality of [n](j,k) equals ceiling(n/k) for j = 1..(n mod k) and equals floor(n/k) for j > (n mod k). Therefore, upon permuting the elements of each [n](j,k) subset, we obtain T(n,k) = (ceiling(n/k)!)^(n mod k)*(floor(n/k)!)^(k-(n mod k)).

Examples

			Triangle begins:
    1;
    2  1;
    6  2 1;
   24  4 2 1;
  120 12 4 2 1;
  ...
T(6,3) counts the 8 permutations of [6] where all cycle-mates are identical mod 3, namely, (1 4)(2 5)(3 6), (1 4)(2 5)(3)(6), (1 4)(2)(5)(3 6), (1)(4)(2 5)(3 6), (1 4)(2)(5)(3)(6), (1)(4)(2 5)(3)(6), (1)(4)(2)(5)(3 6) and (1)(2)(3)(4)(5)(6).
		

Links

• Per Alexandersson, Frether Getachew Kebede, Samuel Asefa Fufa, and Dun Qiu, Pattern-Avoidance and Fuss-Catalan Numbers, J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.

Crossrefs

Cf. A000142, A010551, A264557, A264635, A264656.

Cf. A275062.

Programs

Maple

seq(seq((ceil(n/k)!)^(n mod k)*(floor(n/k)!)^(k-(n mod k)), k=1..n), n=1..10);

Mathematica

Table[(Ceiling[n/k]!)^Mod[n, k]*(Floor[n/k]!)^(k - Mod[n, k]), {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Jun 28 2020 *)

Formula

T(n,k) = (ceiling(n/k)!)^(n mod k)*(floor(n/k)!)^(k-(n mod k)) for 1 <= k <= n.

T(n,1) = A000142(n).

T(n,2) = A010551(n) for n > 1.

T(n,3) = A264557(n) for n > 2.

T(n,4) = A264635(n) for n > 3.

T(n,5) = A264656(n) for n > 4.

T(n,k) = Product_{i=0..k-1} floor((n+i)/k)!. - Alois P. Heinz, May 23 2020