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A326033 Number of knapsack partitions of n such that no addition of one part equal to an existing part is knapsack. 0
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, 1, 1, 0, 8, 0, 8, 4, 3, 0, 11, 5, 3, 4, 5, 0, 30, 2, 9, 9, 20, 3, 37, 6, 18, 16, 37, 20, 71, 12, 37, 40 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,21

COMMENTS

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

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

The partition (10,8,6,6) is counted under a(30) because (10,10,8,6,6), (10,8,8,6,6), and (10,8,6,6,6) are not knapsack.

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]}]])];

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

maxks[n_]:=Select[IntegerPartitions[n], ksQ[#]&&Select[Table[Sort[Append[#, i]], {i, Union[#]}], ksQ]=={}&];

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

CROSSREFS

Cf. A002033, A108917, A275972, A276024, A299702.

Cf. A325857, A325862, A325863, A325864, A325865, A326015, A326016, A326018.

Sequence in context: A001842 A216654 A326016 * A029429 A064559 A340998

Adjacent sequences:  A326030 A326031 A326032 * A326034 A326035 A326036

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jun 03 2019

STATUS

approved

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Last modified May 12 05:25 EDT 2021. Contains 343811 sequences. (Running on oeis4.)