A321736 Number of non-isomorphic weight-n multiset partitions whose part-sizes are also their vertex-degrees.

1, 1, 2, 4, 9, 17, 42, 92, 231, 579, 1577

Offset: 0

Author

Gus Wiseman, Nov 19 2018

Keywords

• nonn
• more

Comments

Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with the same multiset of row sums as of column sums.

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) = 17 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},{1,2,2}}
                    {{2},{1,2}}    {{1,2},{1,2}}      {{1,1},{2,2,2}}
                    {{1},{2},{3}}  {{1},{2,2,2}}      {{1,2},{1,2,2}}
                                   {{2},{1,2,2}}      {{1},{2,2,2,2}}
                                   {{1},{1},{2,3}}    {{2},{1,2,2,2}}
                                   {{1},{2},{3,3}}    {{1},{2,2},{3,3}}
                                   {{1},{3},{2,3}}    {{1},{2,3},{2,3}}
                                   {{1},{2},{3},{4}}  {{1},{2},{3,3,3}}
                                                      {{1},{3},{2,3,3}}
                                                      {{2},{1,2},{3,3}}
                                                      {{2},{1,3},{2,3}}
                                                      {{3},{3},{1,2,3}}
                                                      {{1},{2},{2},{3,4}}
                                                      {{1},{2},{3},{4,4}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{3},{4},{5}}
		

Crossrefs

Cf. A000700, A007716, A057150, A120732, A319056, A319616, A320451, A321719, A321721, A321722, A321724, A321732, A321733, A321735, A321739.