A326035 Number of uniform knapsack partitions of n.

1, 1, 2, 3, 4, 4, 6, 6, 9, 10, 12, 12, 17, 16, 20, 25, 27, 29, 35, 39, 44, 57, 53, 66, 75, 84, 84, 114, 112, 131, 133, 162, 167, 209, 192, 242, 250, 289, 279, 363, 348, 417, 404, 502, 487, 608, 557, 706, 682, 835, 773, 1004, 922, 1149, 1059, 1344, 1257, 1595

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

Cf. A002033, A047966, A072774, A108917, A275972, A276024, A299702.

Cf. A325592, A325858, A326015, A326016, A326017, A326036, A326037.

Sequence in context: A179276 A213635 A218443 * A205787 A072455 A177862

Adjacent sequences: A326032 A326033 A326034 * A326036 A326037 A326038

Keyword

nonn

Author

Gus Wiseman, Jun 04 2019

Status

approved