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 A326015 Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack. 8
 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 4, 4, 5, 3, 3, 4, 6, 2, 7, 6, 13, 9, 19, 16, 27, 21, 40, 33, 47, 37, 54, 48, 66, 51, 65, 65, 77, 64, 80, 71, 96, 60, 106, 95, 112, 93, 152, 114, 191, 131, 242, 192, 303, 210, 366, 300, 482, 352, 581, 450, 713, 539, 882, 689, 995 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS An integer partition is knapsack if every distinct submultiset has a different sum. These are the subsets counted by A325867, ordered by sum rather than maximum. LINKS Fausto A. C. Cariboni, Table of n, a(n) for n = 1..600 EXAMPLE The a(1) = 1 through a(17) = 6 strict knapsack partitions (empty columns not shown):   {1}  {2,1}  {3,1}  {3,2}  {4,2,1}  {5,2,1}  {4,3,2}  {6,3,1}  {5,4,2}                                               {5,3,1}  {7,2,1}  {6,3,2}                                               {6,2,1}           {6,4,1}                                                                 {7,3,1} .   {5,4,3}  {6,4,3}  {6,5,3}  {6,5,4}    {7,5,4}    {7,6,4}   {7,3,2}  {6,5,2}  {8,5,1}  {7,6,2}    {9,4,3}    {9,5,3}   {7,4,1}  {7,4,2}  {9,3,2}  {8,4,2,1}  {9,6,1}    {9,6,2}   {8,3,1}  {7,5,1}                      {9,4,2,1}  {8,4,3,2}            {9,3,1}                                 {9,5,2,1}                                                    {10,4,2,1} MATHEMATICA ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]] maxsks[n_]:=Select[Select[IntegerPartitions[n], UnsameQ@@#&&ksQ[#]&], Select[Table[Append[#, i], {i, Complement[Range[Max@@#], #]}], ksQ]=={}&]; Table[Length[maxsks[n]], {n, 30}] CROSSREFS Cf. A002033, A108917, A275972, A276024. Cf. A325863, A325864, A325877, A325878, A325880, A326016, A326017, A326018. Sequence in context: A322349 A322348 A321232 * A078822 A224980 A154392 Adjacent sequences:  A326012 A326013 A326014 * A326016 A326017 A326018 KEYWORD nonn,changed AUTHOR Gus Wiseman, Jun 03 2019 STATUS approved

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Last modified January 18 07:55 EST 2022. Contains 350454 sequences. (Running on oeis4.)