OFFSET
0,6
LINKS
Sean A. Irvine, Java program (github)
EXAMPLE
The sequence of product-sum knapsack partitions begins:
0: ()
1: (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)
The partition (4,4,3) is not a sum-product knapsack partition of 11 because (4*4) = (4)+(4*3).
A complete list of all sums of products of multiset partitions of submultisets of (5,4,2) is:
0 = 0
(2) = 2
(4) = 4
(5) = 5
(2*4) = 8
(2*5) = 10
(4*5) = 20
(2*4*5) = 40
(2)+(4) = 6
(2)+(5) = 7
(2)+(4*5) = 22
(4)+(5) = 9
(4)+(2*5) = 14
(5)+(2*4) = 13
(2)+(4)+(5) = 11
These are all distinct, so (5,4,2) is a sum-product knapsack partition of 11.
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]]]];
rrtuks[n_]:=Select[IntegerPartitions[n], Function[q, UnsameQ@@Apply[Plus, Apply[Times, Union@@mps/@Union[Subsets[q]], {2}], {1}]]];
Table[Length[rrtuks[n]], {n, 12}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Oct 04 2018
EXTENSIONS
a(13)-a(37) from Sean A. Irvine, Jul 13 2022
STATUS
approved