A275069 Number A(n,k) of set partitions of [n] such that i-j is a multiple of k for all i,j belonging to the same block; 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, 5, 1, 1, 1, 1, 2, 15, 1, 1, 1, 1, 1, 4, 52, 1, 1, 1, 1, 1, 2, 10, 203, 1, 1, 1, 1, 1, 1, 4, 25, 877, 1, 1, 1, 1, 1, 1, 2, 8, 75, 4140, 1, 1, 1, 1, 1, 1, 1, 4, 20, 225, 21147, 1, 1, 1, 1, 1, 1, 1, 2, 8, 50, 780, 115975, 1

Offset: 0

Author

Alois P. Heinz, Jul 15 2016

Keywords

• nonn
• tabl

Examples

			A(5,0) = 1: 1|2|3|4|5.
A(5,1) = 52 = A000110(5).
A(5,2) = 10: 135|24, 13|24|5, 135|2|4, 13|2|4|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
A(5,3) = 4: 14|25|3, 14|2|3|5, 1|25|3|4, 1|2|3|4|5.
A(5,4) = 2: 15|2|3|4, 1|2|3|4|5.
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,      5,    2,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,     15,    4,   2,   1,  1,  1, 1, 1, 1, 1, ...
  1,     52,   10,   4,   2,  1,  1, 1, 1, 1, 1, ...
  1,    203,   25,   8,   4,  2,  1, 1, 1, 1, 1, ...
  1,    877,   75,  20,   8,  4,  2, 1, 1, 1, 1, ...
  1,   4140,  225,  50,  16,  8,  4, 2, 1, 1, 1, ...
  1,  21147,  780, 125,  40, 16,  8, 4, 2, 1, 1, ...
  1, 115975, 2704, 375, 100, 32, 16, 8, 4, 2, 1, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• Wikipedia, Partition of a set

Crossrefs

Columns k=0-10 give: A000012, A000110, A124419, A275070, A275071, A275072, A275073, A275074, A275075, A275076, A275077.

A(k*n,n) for k=1-4 gives: A000012, A000079, A000351, A001024.

Programs

Maple

with(combinat):
A:= (n, k)-> mul(bell(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[BellB[Floor[(n+i)/k]], {i, 0, k-1}]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 17 2017, translated from Maple *)

Formula

A(n,k) = Product_{i=0..k-1} A000110(floor((n+i)/k)).