A321229 Number of non-isomorphic connected weight-n multiset partitions with multiset density -1.

1, 1, 3, 6, 16, 37, 105, 279, 817, 2387, 7269

Offset: 0

Author

Gus Wiseman, Oct 31 2018

Keywords

• nonn
• more

Comments

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 37 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}        {{1,1,2,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,2,2}}        {{1,2,2,2,2}}
                    {{1},{1,1}}    {{1,2,3,3}}        {{1,2,2,3,3}}
                    {{2},{1,2}}    {{1,2,3,4}}        {{1,2,3,3,3}}
                    {{1},{1},{1}}  {{1},{1,1,1}}      {{1,2,3,4,4}}
                                   {{1,1},{1,1}}      {{1,2,3,4,5}}
                                   {{1},{1,2,2}}      {{1},{1,1,1,1}}
                                   {{1,2},{2,2}}      {{1,1},{1,1,1}}
                                   {{1,3},{2,3}}      {{1,1},{1,2,2}}
                                   {{2},{1,2,2}}      {{1},{1,2,2,2}}
                                   {{3},{1,2,3}}      {{1,2},{2,2,2}}
                                   {{1},{1},{1,1}}    {{1,2},{2,3,3}}
                                   {{1},{2},{1,2}}    {{1,3},{2,3,3}}
                                   {{2},{2},{1,2}}    {{1,4},{2,3,4}}
                                   {{1},{1},{1},{1}}  {{2},{1,1,2,2}}
                                                      {{2},{1,2,2,2}}
                                                      {{2},{1,2,3,3}}
                                                      {{2,2},{1,2,2}}
                                                      {{3},{1,2,3,3}}
                                                      {{3,3},{1,2,3}}
                                                      {{4},{1,2,3,4}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1,2,2}}
                                                      {{1},{1,2},{2,2}}
                                                      {{1},{2},{1,2,2}}
                                                      {{2},{1,2},{2,2}}
                                                      {{2},{1,3},{2,3}}
                                                      {{2},{2},{1,2,2}}
                                                      {{2},{3},{1,2,3}}
                                                      {{3},{1,3},{2,3}}
                                                      {{3},{3},{1,2,3}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}
		

Crossrefs

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321227, A321228, A321231.