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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
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}]
CROSSREFS
Row sums are A108917.
Column k = 3 is A326034.
Sequence in context: A321926 A037870 A250205 * A290307 A206588 A302234
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jun 03 2019
STATUS
approved