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A326015 Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack. 7
1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 4, 4, 5, 3, 3, 4, 6, 2, 7, 6, 13, 9, 19, 16, 27, 21, 40, 33, 47, 37, 54, 48, 66, 51, 65, 65, 77, 64, 80, 71, 96, 60, 106, 95, 112, 93, 152, 114, 191, 131, 242, 192, 303, 210, 366, 300, 482, 352, 581, 450, 713, 539, 882, 689, 995 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

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

These are the subsets counted by A325867, ordered by sum rather than maximum.

LINKS

Table of n, a(n) for n=1..65.

EXAMPLE

The a(1) = 1 through a(17) = 6 strict knapsack partitions (empty columns not shown):

  {1}  {2,1}  {3,1}  {3,2}  {4,2,1}  {5,2,1}  {4,3,2}  {6,3,1}  {5,4,2}

                                              {5,3,1}  {7,2,1}  {6,3,2}

                                              {6,2,1}           {6,4,1}

                                                                {7,3,1}

.

  {5,4,3}  {6,4,3}  {6,5,3}  {6,5,4}    {7,5,4}    {7,6,4}

  {7,3,2}  {6,5,2}  {8,5,1}  {7,6,2}    {9,4,3}    {9,5,3}

  {7,4,1}  {7,4,2}  {9,3,2}  {8,4,2,1}  {9,6,1}    {9,6,2}

  {8,3,1}  {7,5,1}                      {9,4,2,1}  {8,4,3,2}

           {9,3,1}                                 {9,5,2,1}

                                                   {10,4,2,1}

MATHEMATICA

ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]]

maxsks[n_]:=Select[Select[IntegerPartitions[n], UnsameQ@@#&&ksQ[#]&], Select[Table[Append[#, i], {i, Complement[Range[Max@@#], #]}], ksQ]=={}&];

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

CROSSREFS

Cf. A002033, A108917, A275972, A276024.

Cf. A325863, A325864, A325877, A325878, A325880, A326016, A326017, A326018.

Sequence in context: A322349 A322348 A321232 * A078822 A224980 A154392

Adjacent sequences:  A326012 A326013 A326014 * A326016 A326017 A326018

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 03 2019

STATUS

approved

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Last modified February 28 00:28 EST 2020. Contains 332319 sequences. (Running on oeis4.)