A274859 Number A(n,k) of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 8, 1, 1, 2, 4, 15, 16, 1, 1, 2, 4, 8, 52, 32, 1, 1, 2, 4, 8, 18, 203, 64, 1, 1, 2, 4, 8, 16, 40, 877, 128, 1, 1, 2, 4, 8, 16, 32, 101, 4140, 256, 1, 1, 2, 4, 8, 16, 32, 68, 254, 21147, 512, 1, 1, 2, 4, 8, 16, 32, 64, 144, 723, 115975, 1024

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..34, flattened

Example

Square array A(n,k) begins:
:   1,      1,    1,   1,   1,   1,   1, ...
:   1,      1,    1,   1,   1,   1,   1, ...
:   2,      2,    2,   2,   2,   2,   2, ...
:   4,      5,    4,   4,   4,   4,   4, ...
:   8,     15,    8,   8,   8,   8,   8, ...
:  16,     52,   18,  16,  16,  16,  16, ...
:  32,    203,   40,  32,  32,  32,  32, ...
:  64,    877,  101,  68,  64,  64,  64, ...
: 128,   4140,  254, 144, 128, 128, 128, ...
: 256,  21147,  723, 304, 264, 256, 256, ...
: 512, 115975, 2064, 692, 544, 512, 512, ...

Maple

b:= proc(l, k, i, t) option remember; `if`(l=[], 1, add(`if`(l[j]=t,
      b(subsop(j=[][], l), k, j, irem(1+t, k)), 0), j=[1, $i..nops(l)]))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k=0, 2^(n-1), b([seq(
            irem(i, k), i=2..n)], k, 1, irem(2, k)))):
seq(seq(A(n, d-n), n=0..d), d=0..15);

Mathematica

b[l_, k_, i_, t_] := b[l, k, i, t] = If[l == {}, 1, Sum[If[l[[j]] == t, b[ReplacePart[l, j -> Nothing], k, j, Mod[1+t, k]], 0], {j, Prepend[ Range[i, Length[l]], 1]}]]; A[n_, k_] := If[n==0, 1, If[k==0, 2^(n-1), b[Flatten[Table[Mod[i, k], {i, 2, n}]], k, 1, Mod[2, k]]]]; Table[A[n, d - n], {d, 0, 15}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 02 2017, translated from Maple *)

Crossrefs

Columns k=0-10 give: A011782, A000110, A274547, A274860, A274861, A274862, A274863, A274864, A274865, A274866, A274867.

Sequence in context: A140993 A027935 A137940 * A137855 A113143 A181802

Adjacent sequences: A274856 A274857 A274858 * A274860 A274861 A274862

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 09 2016

Status

approved