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

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

Offset

0,3

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.

Links

Table of n, a(n) for n=0..6.

Example

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.

Sequence in context: A364393 A324311 A303671 * A370766 A102087 A372234

Adjacent sequences: A330660 A330661 A330662 * A330664 A330665 A330666

Keyword

nonn,more

Author

Gus Wiseman, Dec 27 2019

Status

approved