OFFSET
0,19
COMMENTS
An integer partition is knapsack if every distinct submultiset has a different sum.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010
Fausto A. C. Cariboni, Conjectures on columns of T(n,k), Jun 05 2021.
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 1 1 1
0 1 1 1 1
0 1 1 2 1 1
0 1 1 1 2 1 1
0 1 1 2 3 2 1 1
0 1 1 2 1 3 2 1 1
0 1 1 2 2 4 3 2 1 1
0 1 1 2 3 1 4 3 2 1 1
0 1 1 3 3 4 6 4 3 2 1 1
0 1 1 1 1 3 1 6 4 3 2 1 1
0 1 1 3 3 5 4 7 6 4 3 2 1 1
0 1 1 2 3 5 4 1 7 6 4 3 2 1 1
0 1 1 2 3 4 6 6 11 7 6 4 3 2 1 1
Row n = 9 counts the following partitions:
(111111111) (22221) (333) (432) (54) (63) (72) (81) (9)
(3222) (441) (522) (621) (711)
(531) (6111)
(51111)
MATHEMATICA
ks[n_]:=Select[IntegerPartitions[n], UnsameQ@@Total/@Union[Subsets[#]]&];
Table[Length[Select[ks[n], Length[#]==k==0||Max@@#==k&]], {n, 0, 15}, {k, 0, n}]
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jun 03 2019
STATUS
approved