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

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

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..10.

Example

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.

Sequence in context: A264564 A304963 A034730 * A095127 A006342 A258041

Adjacent sequences: A321140 A321141 A321142 * A321144 A321145 A321146

Keyword

nonn,more

Author

Gus Wiseman, Oct 28 2018

Status

approved