|
|
A326017
|
|
Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.
|
|
11
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,19
|
|
COMMENTS
|
An integer partition is knapsack if every distinct submultiset has a different sum.
|
|
LINKS
|
|
|
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}]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|