Revision History for A326033
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing all changes.
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#7 by Susanna Cuyler at Tue Jun 04 08:37:03 EDT 2019
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#6 by Gus Wiseman at Mon Jun 03 23:47:17 EDT 2019
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#5 by Gus Wiseman at Mon Jun 03 23:45:44 EDT 2019
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| MATHEMATICA
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ks[n_]:=Select[IntegerPartitions[n], Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
maxks[n_]:=Select[ksIntegerPartitions[n], ksQ[#]&&Select[Table[Sort[Append[#, i]], {i, Union[#]}], ksQ]=={}&];
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#4 by Gus Wiseman at Mon Jun 03 23:42:49 EDT 2019
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| EXAMPLE
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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.
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#3 by Gus Wiseman at Mon Jun 03 05:49:00 EDT 2019
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| NAME
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allocatedNumber of knapsack partitions of n such that no addition of one part equal to an existing forpart Gusis Wisemanknapsack.
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| DATA
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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
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| OFFSET
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1,21
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| COMMENTS
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An integer partition is knapsack if every distinct submultiset has a different sum.
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| EXAMPLE
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The partition (10,8,6,6) is counted under a(30).
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| MATHEMATICA
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sums[ptn_]:=sums[ptn]=If[Length[ptn]==1, ptn, Union@@(Join[sums[#], sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn, i], {i, Length[ptn]}]])];
ks[n_]:=Select[IntegerPartitions[n], Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;
maxks[n_]:=Select[ks[n], Select[Table[Sort[Append[#, i]], {i, Union[#]}], ksQ]=={}&];
Table[Length[maxks[n]], {n, 30}]
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| CROSSREFS
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Cf. A002033, A108917, A275972, A276024, A299702.
Cf. A325857, A325862, A325863, A325864, A325865, A326015, A326016, A326018.
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| KEYWORD
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allocated
nonn,more
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| AUTHOR
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Gus Wiseman, Jun 03 2019
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| STATUS
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approved
editing
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#2 by Gus Wiseman at Mon Jun 03 05:23:40 EDT 2019
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| KEYWORD
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allocating
allocated
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#1 by Gus Wiseman at Mon Jun 03 05:23:40 EDT 2019
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| NAME
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allocated for Gus Wiseman
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| KEYWORD
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allocating
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| STATUS
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approved
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