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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

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

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

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

Cf. A267597, A320052, A320054, A320055, A320056, A320057, A320058.

Sequence in context: A275727 A255395 A175266 * A098235 A342847 A345873

Adjacent sequences: A320050 A320051 A320052 * A320054 A320055 A320056

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Oct 04 2018

STATUS

approved

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Last modified January 31 11:22 EST 2023. Contains 359971 sequences. (Running on oeis4.)