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 A367215 Number of strict integer partitions of n whose length (number of parts) is not equal to the sum of any subset. 24
 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 5, 7, 8, 10, 12, 15, 18, 21, 25, 29, 34, 40, 46, 53, 62, 71, 82, 95, 109, 124, 143, 162, 185, 210, 240, 270, 308, 347, 393, 443, 500, 562, 634, 711, 798, 895, 1002, 1120, 1252, 1397, 1558, 1735, 1930, 2146, 2383, 2644, 2930, 3245 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS These partitions have Heinz numbers A367225 /\ A005117. LINKS Chai Wah Wu, Table of n, a(n) for n = 0..118 EXAMPLE The a(2) = 1 through a(11) = 7 strict partitions: (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (3,1) (4,1) (5,1) (4,3) (5,3) (5,4) (6,4) (6,5) (6,1) (7,1) (6,3) (7,3) (7,4) (8,1) (9,1) (8,3) (5,4,1) (10,1) (5,4,2) (6,4,1) The a(2) = 1 through a(15) = 15 strict partitions (A..F = 10..15): 2 3 4 5 6 7 8 9 A B C D E F 31 41 51 43 53 54 64 65 75 76 86 87 61 71 63 73 74 84 85 95 96 81 91 83 93 94 A4 A5 541 A1 B1 A3 B3 B4 542 642 C1 D1 C3 641 651 652 752 E1 741 742 761 654 751 842 762 841 851 852 941 861 6521 942 951 A41 7521 MATHEMATICA Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#], Length[#]]&]], {n, 0, 30}] CROSSREFS The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred. sum-full sum-free comb-full comb-free ------------------------------------------- partitions: A367212 A367213 A367218 A367219 strict: A367214 A367215* A367220 A367221 subsets: A367216 A367217 A367222 A367223 ranks: A367224 A367225 A367226 A367227 A000041 counts integer partitions, strict A000009. A007865/A085489/A151897 count certain types of sum-free subsets. A124506 appears to count combination-free subsets, differences of A326083. A188431 counts complete strict partitions, incomplete A365831. A237667 counts sum-free partitions, ranks A364531. A240861 counts strict partitions with length not a part, complement A240855. A275972 counts strict knapsack partitions, non-strict A108917. A364349 counts sum-free strict partitions, sum-full A364272. Triangles: A008289 counts strict partitions by length, non-strict A008284. A365661 counts strict partitions with a subset-sum k, non-strict A365543. A365663 counts strict partitions without a subset-sum k, non-strict A046663. A365832 counts strict partitions by subset-sums, non-strict A365658. Cf. A002865, A229816, A238628, A364346, A364350, A365312, A365922, A366320. Sequence in context: A074732 A089046 A054911 * A137267 A341451 A123576 Adjacent sequences: A367212 A367213 A367214 * A367216 A367217 A367218 KEYWORD nonn AUTHOR Gus Wiseman, Nov 12 2023 STATUS approved

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Last modified April 18 22:18 EDT 2024. Contains 371782 sequences. (Running on oeis4.)