OFFSET
0,3
COMMENTS
An integer partition is uniform if all parts appear with the same multiplicity, and knapsack if every distinct submultiset has a different sum.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..650
EXAMPLE
The a(1) = 1 through a(8) = 9 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (11111) (51) (61) (62)
(222) (421) (71)
(111111) (1111111) (521)
(2222)
(3311)
(11111111)
MATHEMATICA
sums[ptn_]:=sums[ptn]=If[Length[ptn]==1, ptn, Union@@(Join[sums[#], sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn, i], {i, Length[ptn]}]])];
ks[n_]:=Select[IntegerPartitions[n], Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
Table[Length[Select[ks[n], SameQ@@Length/@Split[#]&]], {n, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 04 2019
STATUS
approved