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A325863 Number of integer partitions of n such that every distinct non-singleton submultiset has a different sum. 7
1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 24, 29, 31, 41, 51, 58, 67, 84, 91, 117, 117 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A knapsack partition (A108917, A299702) is an integer partition such that every submultiset has a different sum. The one non-knapsack partition counted under a(4) is (2,1,1).

LINKS

Table of n, a(n) for n=0..20.

EXAMPLE

The partition (2,1,1,1) has non-singleton submultisets {1,2} and {1,1,1} with the same sum, so (2,1,1,1) is not counted under a(5).

The a(1) = 1 through a(8) = 15 partitions:

  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

       (11)  (21)   (22)    (32)     (33)      (43)       (44)

             (111)  (31)    (41)     (42)      (52)       (53)

                    (211)   (221)    (51)      (61)       (62)

                    (1111)  (311)    (222)     (322)      (71)

                            (11111)  (321)     (331)      (332)

                                     (411)     (421)      (422)

                                     (3111)    (511)      (431)

                                     (111111)  (2221)     (521)

                                               (4111)     (611)

                                               (1111111)  (2222)

                                                          (3311)

                                                          (5111)

                                                          (41111)

                                                          (11111111)

The 10 non-knapsack partitions counted under a(12):

  (7,6,1)

  (7,5,2)

  (7,4,3)

  (7,5,1,1)

  (7,4,2,1)

  (7,3,3,1)

  (7,3,2,2)

  (7,4,1,1,1)

  (7,2,2,2,1)

  (7,1,1,1,1,1,1,1)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], UnsameQ@@Plus@@@Union[Subsets[#, {2, Length[#]}]]&]], {n, 0, 15}]

CROSSREFS

Dominates A108917.

Cf. A002033, A055212, A143823, A196723, A276024, A299702, A325856, A325862, A325864, A325865, A325866, A325867, A325877.

Sequence in context: A239010 A104738 A319469 * A028309 A242717 A026810

Adjacent sequences:  A325860 A325861 A325862 * A325864 A325865 A325866

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, May 31 2019

STATUS

approved

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Last modified May 29 20:42 EDT 2020. Contains 334710 sequences. (Running on oeis4.)