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A326035 Number of uniform knapsack partitions of n. 3
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..57.

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

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Last modified July 4 16:24 EDT 2020. Contains 335448 sequences. (Running on oeis4.)