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Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.
10

%I #7 Jun 04 2019 08:36:02

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,1,0,3,0,0,0,1,0,8,0,

%T 8,4,3,0,11,5,3,2,5,0,29,2,9,8,20,2

%N Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.

%C An integer partition is knapsack if every distinct submultiset has a different sum.

%C The Heinz numbers of these partitions are given by A326018.

%e The initial terms count the following partitions:

%e 15: (5,4,3,3)

%e 21: (7,6,5,3)

%e 21: (7,5,3,3,3)

%e 24: (8,7,6,3)

%e 25: (7,5,5,4,4)

%e 27: (9,8,7,3)

%e 27: (9,7,6,5)

%e 27: (8,7,3,3,3,3)

%e 31: (10,8,6,6,1)

%e 33: (11,9,7,3,3)

%e 33: (11,8,5,5,4)

%e 33: (11,7,6,6,3)

%e 33: (11,7,3,3,3,3,3)

%e 33: (11,5,5,4,4,4)

%e 33: (10,9,8,3,3)

%e 33: (10,8,6,6,3)

%e 33: (10,8,3,3,3,3,3)

%t sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];

%t ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;

%t maxks[n_]:=Select[IntegerPartitions[n],ksQ[#]&&Select[Table[Sort[Append[#,i]],{i,Range[Max@@#]}],ksQ]=={}&];

%t Table[Length[maxks[n]],{n,30}]

%Y Cf. A002033, A108917, A275972, A276024.

%Y Cf. A325863, A325864, A325877, A325878, A325880, A326015, A326017, A326018.

%K nonn,more

%O 1,21

%A _Gus Wiseman_, Jun 03 2019