A326016 Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.

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

Offset

1,21

Comments

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

The Heinz numbers of these partitions are given by A326018.

Links

Table of n, a(n) for n=1..50.

Example

The initial terms count the following partitions:
  15: (5,4,3,3)
  21: (7,6,5,3)
  21: (7,5,3,3,3)
  24: (8,7,6,3)
  25: (7,5,5,4,4)
  27: (9,8,7,3)
  27: (9,7,6,5)
  27: (8,7,3,3,3,3)
  31: (10,8,6,6,1)
  33: (11,9,7,3,3)
  33: (11,8,5,5,4)
  33: (11,7,6,6,3)
  33: (11,7,3,3,3,3,3)
  33: (11,5,5,4,4,4)
  33: (10,9,8,3,3)
  33: (10,8,6,6,3)
  33: (10,8,3,3,3,3,3)

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, Range[Max@@#]}], ksQ]=={}&];
Table[Length[maxks[n]], {n, 30}]

Crossrefs

Cf. A002033, A108917, A275972, A276024.

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

Sequence in context: A001842 A216654 A364014 * A326033 A029429 A064559

Adjacent sequences: A326013 A326014 A326015 * A326017 A326018 A326019

Keyword

nonn,more

Author

Gus Wiseman, Jun 03 2019

Status

approved