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 A267597 Number of 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 any submultiset of y is distinct. 8
 1, 1, 1, 1, 1, 2, 3, 3, 4, 4, 6, 7, 8, 12, 12, 14, 18, 23, 23, 32, 30, 35, 50, 48, 47, 56, 80, 77, 87, 105, 100, 134, 139, 145, 194, 170, 192, 250 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS Table of n, a(n) for n=0..37. 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 Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913. Cf. A320052, A320053, A320054, A320055, A320056, A320057, A320058. Sequence in context: A230476 A103297 A274017 * A238221 A320052 A168173 Adjacent sequences: A267594 A267595 A267596 * A267598 A267599 A267600 KEYWORD nonn,more AUTHOR Gus Wiseman, Oct 04 2018 EXTENSIONS a(13)-a(37) from Sean A. Irvine, Jul 13 2022 STATUS approved

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Last modified April 19 02:45 EDT 2024. Contains 371782 sequences. (Running on oeis4.)