A330663 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth.

1, 1, 2, 4, 20, 140, 1411

Offset: 0

Author

Gus Wiseman, Dec 27 2019

Keywords

• nonn
• more

Comments

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

Examples

			Non-isomorphic representatives of the a(2) = 2 through a(4) = 20 multisystems:
  {1,1}  {{1},{1,1}}  {{{1}},{{1},{1,1}}}
  {1,2}  {{1},{1,2}}  {{{1,1}},{{1},{1}}}
         {{1},{2,3}}  {{{1}},{{1},{1,2}}}
         {{2},{1,1}}  {{{1,1}},{{1},{2}}}
                      {{{1}},{{1},{2,2}}}
                      {{{1,1}},{{2},{2}}}
                      {{{1}},{{1},{2,3}}}
                      {{{1,1}},{{2},{3}}}
                      {{{1}},{{2},{1,1}}}
                      {{{1,2}},{{1},{1}}}
                      {{{1}},{{2},{1,2}}}
                      {{{1,2}},{{1},{2}}}
                      {{{1}},{{2},{1,3}}}
                      {{{1,2}},{{1},{3}}}
                      {{{1}},{{2},{3,4}}}
                      {{{1,2}},{{3},{4}}}
                      {{{2}},{{1},{1,1}}}
                      {{{2}},{{1},{1,3}}}
                      {{{2}},{{3},{1,1}}}
                      {{{2,3}},{{1},{1}}}
		

Crossrefs

The non-maximal version is A330474.

Labeled versions are A330675 (strongly normal) and A330676 (normal).

The case where the leaves are sets (as opposed to multisets) is A330677.

The case with all atoms distinct is A000111.

The case with all atoms equal is (also) A000111.

Cf. A000311, A004114, A005121, A006472, A007716, A048816, A141268, A306186, A330470, A330655, A330664.