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%I #6 Jun 02 2019 00:50:11
%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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