A274835 Number A(n,k) of set partitions of [n] such that the difference between each element and its block index is a multiple of k; 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, 3, 52, 1, 1, 1, 1, 1, 2, 7, 203, 1, 1, 1, 1, 1, 1, 3, 14, 877, 1, 1, 1, 1, 1, 1, 2, 4, 39, 4140, 1, 1, 1, 1, 1, 1, 1, 3, 9, 95, 21147, 1, 1, 1, 1, 1, 1, 1, 2, 4, 18, 304, 115975, 1

Offset

0,9

Links

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

Wikipedia, Partition of a set

Example

A(3,0) = 1: 1|2|3.
A(3,1) = 5: 123, 12|3, 13|2, 1|23, 1|2|3.
A(5,2) = 7: 135|24, 13|24|5, 15|24|3, 1|24|35, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
A(7,3) = 9: 147|25|36, 14|25|36|7, 17|25|36|4, 1|25|36|47, 17|2|36|4|5, 1|2|36|47|5, 17|2|3|4|5|6, 1|2|3|47|5|6, 1|2|3|4|5|6|7.
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,   3,  2,  1, 1, 1, 1, 1, 1, 1, ...
  1,     52,   7,  3,  2, 1, 1, 1, 1, 1, 1, ...
  1,    203,  14,  4,  3, 2, 1, 1, 1, 1, 1, ...
  1,    877,  39,  9,  4, 3, 2, 1, 1, 1, 1, ...
  1,   4140,  95, 18,  5, 4, 3, 2, 1, 1, 1, ...
  1,  21147, 304, 33, 11, 5, 4, 3, 2, 1, 1, ...
  1, 115975, 865, 89, 22, 6, 5, 4, 3, 2, 1, ...

Maple

b:= proc(n, k, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, k)=0, b(n-1, k, max(m, j),
              irem(t+1, k)), 0), j=1..m+1))
    end:
A:= (n, k)-> `if`(k=0, 1, b(n, k, 0, 1)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

b[n_, k_, m_, t_] := b[n, k, m, t] = If[n==0, 1, Sum[If[Mod[j-t, k]==0, b[n-1, k, Max[m, j], Mod[t+1, k]], 0], {j, 1, m+1}]]; A[n_, k_]:= If[k==0, 1, b[n, k, 0, 1]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000012, A000110, A274538, A274836, A274837, A274838, A274839, A274840, A274841, A274842, A274843.

Main diagonal gives A000012.

A(n,ceiling(n/2)) gives A008619.

A(3n,n) gives A094002.

Sequence in context: A333418 A212363 A212382 * A275069 A181937 A233836

Adjacent sequences: A274832 A274833 A274834 * A274836 A274837 A274838

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 08 2016

Status

approved