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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
- Wikipedia, Partition of a set
Crossrefs
Programs
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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 *)