A321143 Number of non-isomorphic knapsack multiset partitions of weight n.

1, 1, 4, 10, 31, 87, 272, 835, 2673, 8805, 29583

Offset: 0

Author

Gus Wiseman, Oct 28 2018

Keywords

• nonn
• more

Comments

A multiset partition is knapsack if every distinct submultiset of the parts has a different multiset union.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 31 knapsack multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,2,2}}
         {{1},{2}}  {{1},{1,1}}    {{1,2,3,3}}
                    {{1},{2,2}}    {{1,2,3,4}}
                    {{1},{2,3}}    {{1},{1,1,1}}
                    {{2},{1,2}}    {{1,1},{1,1}}
                    {{1},{1},{1}}  {{1},{1,2,2}}
                    {{1},{2},{2}}  {{1,1},{2,2}}
                    {{1},{2},{3}}  {{1,2},{1,2}}
                                   {{1},{2,2,2}}
                                   {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}
Missing from this list are {{1},{1},{1,1}} and {{1},{2},{1,2}}, which are not knapsack.
		

Crossrefs

Cf. A002219, A006827, A007716, A108917, A275972, A276024, A292886, A316983, A319616.