A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

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.

Cf. A002033, A196723, A275972, A276024.

Cf. A325592, A325857, A325862, A325863, A325864, A325877, A326016, A326018.

Sequence in context: A321926 A037870 A250205 * A290307 A206588 A302234

Adjacent sequences: A326014 A326015 A326016 * A326018 A326019 A326020

Keyword

nonn,tabl

Author

Gus Wiseman, Jun 03 2019

Status

approved