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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.

%I A274859 #11 Feb 02 2017 09:32:31
%S A274859 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,
%T A274859 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,
%U A274859 2,4,8,16,32,68,254,21147,512,1,1,2,4,8,16,32,64,144,723,115975,1024
%N 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.
%H A274859 Alois P. Heinz, <a href="/A274859/b274859.txt">Antidiagonals n = 0..34, flattened</a>
%e A274859 Square array A(n,k) begins:
%e A274859 :   1,      1,    1,   1,   1,   1,   1, ...
%e A274859 :   1,      1,    1,   1,   1,   1,   1, ...
%e A274859 :   2,      2,    2,   2,   2,   2,   2, ...
%e A274859 :   4,      5,    4,   4,   4,   4,   4, ...
%e A274859 :   8,     15,    8,   8,   8,   8,   8, ...
%e A274859 :  16,     52,   18,  16,  16,  16,  16, ...
%e A274859 :  32,    203,   40,  32,  32,  32,  32, ...
%e A274859 :  64,    877,  101,  68,  64,  64,  64, ...
%e A274859 : 128,   4140,  254, 144, 128, 128, 128, ...
%e A274859 : 256,  21147,  723, 304, 264, 256, 256, ...
%e A274859 : 512, 115975, 2064, 692, 544, 512, 512, ...
%p A274859 b:= proc(l, k, i, t) option remember; `if`(l=[], 1, add(`if`(l[j]=t,
%p A274859       b(subsop(j=[][], l), k, j, irem(1+t, k)), 0), j=[1, $i..nops(l)]))
%p A274859     end:
%p A274859 A:= (n, k)-> `if`(n=0, 1, `if`(k=0, 2^(n-1), b([seq(
%p A274859             irem(i, k), i=2..n)], k, 1, irem(2, k)))):
%p A274859 seq(seq(A(n, d-n), n=0..d), d=0..15);
%t A274859 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 *)
%Y A274859 Columns k=0-10 give: A011782, A000110, A274547, A274860, A274861, A274862, A274863, A274864, A274865, A274866, A274867.
%K A274859 nonn,tabl
%O A274859 0,6
%A A274859 _Alois P. Heinz_, Jul 09 2016