A320052 Number of product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums of the parts of a multiset partition of any submultiset of y is distinct.

1, 0, 1, 1, 1, 2, 3, 3, 4, 4, 6, 8, 8

Offset

0,6

Links

Table of n, a(n) for n=0..12.

Example

The sequence of product-sum knapsack partitions begins:
   0: ()
   1:
   2: (2)
   3: (3)
   4: (4)
   5: (5) (3,2)
   6: (6) (4,2) (3,3)
   7: (7) (5,2) (4,3)
   8: (8) (6,2) (5,3) (4,4)
   9: (9) (7,2) (6,3) (5,4)
  10: (10) (8,2) (7,3) (6,4) (5,5) (4,3,3)
  11: (11) (9,2) (8,3) (7,4) (6,5) (5,4,2) (5,3,3) (4,4,3)
  12: (12) (10,2) (9,3) (8,4) (7,5) (7,3,2) (6,6) (4,4,4)
A complete list of all products of sums of multiset partitions of submultisets of (4,3,3) is:
           () = 1
          (3) = 3
          (4) = 4
        (3+3) = 6
        (3+4) = 7
      (3+3+4) = 10
      (3)*(3) = 9
      (3)*(4) = 12
    (3)*(3+4) = 21
    (4)*(3+3) = 24
  (3)*(3)*(4) = 36
These are all distinct, so (4,3,3) is a product-sum knapsack partition of 10.

Mathematica

sps[{}]:={{}};
sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
rrsuks[n_]:=Select[IntegerPartitions[n], Function[q, UnsameQ@@Apply[Times, Apply[Plus, Union@@mps/@Union[Subsets[q]], {2}], {1}]]];
Table[Length[rrsuks[n]], {n, 12}]

Crossrefs

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A267597, A320053, A320054, A320055, A320056, A320057, A320058.

Sequence in context: A274017 A267597 A238221 * A168173 A095916 A130121

Adjacent sequences: A320049 A320050 A320051 * A320053 A320054 A320055

Keyword

nonn,more

Author

Gus Wiseman, Oct 04 2018

Status

approved