A326015 Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack.

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

Offset: 1

Author

Gus Wiseman, Jun 03 2019

Keywords

• nonn

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.

Examples

			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}
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 1..600

Crossrefs

Cf. A002033, A108917, A275972, A276024.

Cf. A325863, A325864, A325877, A325878, A325880, A326016, A326017, A326018.

Programs

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}]