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A325863 Number of integer partitions of n such that every distinct non-singleton submultiset has a different sum. 7

%I

%S 1,1,2,3,5,6,9,11,15,17,24,29,31,41,51,58,67,84,91,117,117

%N Number of integer partitions of n such that every distinct non-singleton submultiset has a different sum.

%C 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).

%e 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).

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

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

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

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

%e (211) (221) (51) (61) (62)

%e (1111) (311) (222) (322) (71)

%e (11111) (321) (331) (332)

%e (411) (421) (422)

%e (3111) (511) (431)

%e (111111) (2221) (521)

%e (4111) (611)

%e (1111111) (2222)

%e (3311)

%e (5111)

%e (41111)

%e (11111111)

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

%e (7,6,1)

%e (7,5,2)

%e (7,4,3)

%e (7,5,1,1)

%e (7,4,2,1)

%e (7,3,3,1)

%e (7,3,2,2)

%e (7,4,1,1,1)

%e (7,2,2,2,1)

%e (7,1,1,1,1,1,1,1)

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

%Y Dominates A108917.

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

%K nonn,more

%O 0,3

%A _Gus Wiseman_, May 31 2019

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Last modified July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)