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

0,9

Links

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

Wikipedia, Partition of a set

Formula

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

Example

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, ...

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 *)

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.

Sequence in context: A212363 A212382 A274835 * A181937 A233836 A214719

Adjacent sequences: A275066 A275067 A275068 * A275070 A275071 A275072

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 15 2016

Status

approved