cp's OEIS Frontend

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.

%I A274835 #19 Oct 18 2018 16:55:11
%S A274835 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,
%T A274835 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,
%U A274835 95,21147,1,1,1,1,1,1,1,2,4,18,304,115975,1
%N 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.
%H A274835 Alois P. Heinz, <a href="/A274835/b274835.txt">Antidiagonals n = 0..140, flattened</a>
%H A274835 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A274835 A(3,0) = 1: 1|2|3.
%e A274835 A(3,1) = 5: 123, 12|3, 13|2, 1|23, 1|2|3.
%e A274835 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.
%e A274835 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.
%e A274835 Square array A(n,k) begins:
%e A274835   1,      1,   1,  1,  1, 1, 1, 1, 1, 1, 1, ...
%e A274835   1,      1,   1,  1,  1, 1, 1, 1, 1, 1, 1, ...
%e A274835   1,      2,   1,  1,  1, 1, 1, 1, 1, 1, 1, ...
%e A274835   1,      5,   2,  1,  1, 1, 1, 1, 1, 1, 1, ...
%e A274835   1,     15,   3,  2,  1, 1, 1, 1, 1, 1, 1, ...
%e A274835   1,     52,   7,  3,  2, 1, 1, 1, 1, 1, 1, ...
%e A274835   1,    203,  14,  4,  3, 2, 1, 1, 1, 1, 1, ...
%e A274835   1,    877,  39,  9,  4, 3, 2, 1, 1, 1, 1, ...
%e A274835   1,   4140,  95, 18,  5, 4, 3, 2, 1, 1, 1, ...
%e A274835   1,  21147, 304, 33, 11, 5, 4, 3, 2, 1, 1, ...
%e A274835   1, 115975, 865, 89, 22, 6, 5, 4, 3, 2, 1, ...
%p A274835 b:= proc(n, k, m, t) option remember; `if`(n=0, 1,
%p A274835      add(`if`(irem(j-t, k)=0, b(n-1, k, max(m, j),
%p A274835               irem(t+1, k)), 0), j=1..m+1))
%p A274835     end:
%p A274835 A:= (n, k)-> `if`(k=0, 1, b(n, k, 0, 1)):
%p A274835 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A274835 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_ *)
%Y A274835 Columns k=0-10 give: A000012, A000110, A274538, A274836, A274837, A274838, A274839, A274840, A274841, A274842, A274843.
%Y A274835 Main diagonal gives A000012.
%Y A274835 A(n,ceiling(n/2)) gives A008619.
%Y A274835 A(3n,n) gives A094002.
%K A274835 nonn,tabl
%O A274835 0,9
%A A274835 _Alois P. Heinz_, Jul 08 2016