OFFSET
0,9
LINKS
Alois P. Heinz, Antidiagonals n = 0..34, flattened
Wikipedia, Partition of a set
FORMULA
A(n,k) = Sum_{j=0..k} A287640(n,j).
EXAMPLE
A(5,0) = 1: 12345.
A(5,1) = 42 = 52 - 10 = A000110(5) - 10 counts all set partitions of [5] except: 124|3|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|35, 14|2|3|5, 1|24|3|5, 134|2|5.
A(5,2) = 51 = 52 - 1 = A000110(5) - 1 counts all set partitions of [5] except: 134|2|5.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, 2, 2, ...
1, 5, 5, 5, 5, 5, 5, 5, ...
1, 14, 15, 15, 15, 15, 15, 15, ...
1, 42, 51, 52, 52, 52, 52, 52, ...
1, 132, 191, 202, 203, 203, 203, 203, ...
1, 429, 773, 861, 876, 877, 877, 877, ...
MAPLE
b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
[seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
b[0, _] = 1; b[n_, l_List] := b[n, l] = Sum[b[n - 1, Append[ Table[ Max[ l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}];
A[n_, k_] := If[k == 0, 1, b[n, Table[0, k]]];
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 28 2017
STATUS
approved