login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A320053 Number of spanning sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of y is distinct. 7
1, 1, 2, 3, 2, 3, 4, 5, 6, 8, 9, 12, 14 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
EXAMPLE
The sequence of spanning sum-product knapsack partitions begins:
0: ()
1: (1)
2: (2) (1,1)
3: (3) (2,1) (1,1,1)
4: (4) (3,1)
5: (5) (4,1) (3,2)
6: (6) (5,1) (4,2) (3,3)
7: (7) (6,1) (5,2) (4,3) (3,3,1)
8: (8) (7,1) (6,2) (5,3) (4,4) (3,3,2)
9: (9) (8,1) (7,2) (6,3) (5,4) (4,4,1) (4,3,2) (3,3,3)
A complete list of all sums of products covering the parts of (3,3,3,2) is:
(2*3*3*3) = 54
(2)+(3*3*3) = 29
(3)+(2*3*3) = 21
(2*3)+(3*3) = 15
(2)+(3)+(3*3) = 14
(3)+(3)+(2*3) = 12
(2)+(3)+(3)+(3) = 11
These are all distinct, so (3,3,3,2) is a spanning sum-product knapsack partition of 11.
An example of a spanning sum-product knapsack partition that is not a spanning product-sum knapsack partition is (5,4,3,2).
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]]]];
rtuks[n_]:=Select[IntegerPartitions[n], Function[q, UnsameQ@@Apply[Plus, Apply[Times, mps[q], {2}], {1}]]];
Table[Length[rtuks[n]], {n, 8}]
CROSSREFS
Sequence in context: A275727 A255395 A175266 * A098235 A342847 A345873
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Oct 04 2018
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 23 06:44 EDT 2024. Contains 374544 sequences. (Running on oeis4.)