A330474 Number of non-isomorphic balanced reduced multisystems of weight n.

1, 1, 2, 7, 48, 424

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..5.

Example

Non-isomorphic representatives of the a(3) = 7 multisystems:
  {1,1,1}
  {1,1,2}
  {1,2,3}
  {{1},{1,1}}
  {{1},{1,2}}
  {{1},{2,3}}
  {{2},{1,1}}
Non-isomorphic representatives of the a(4) = 48 multisystems:
  {1,1,1,1}  {{1},{1,1,1}}    {{{1}},{{1},{1,1}}}
  {1,1,1,2}  {{1,1},{1,1}}    {{{1,1}},{{1},{1}}}
  {1,1,2,2}  {{1},{1,1,2}}    {{{1}},{{1},{1,2}}}
  {1,1,2,3}  {{1,1},{1,2}}    {{{1,1}},{{1},{2}}}
  {1,2,3,4}  {{1},{1,2,2}}    {{{1}},{{1},{2,2}}}
             {{1,1},{2,2}}    {{{1,1}},{{2},{2}}}
             {{1},{1,2,3}}    {{{1}},{{1},{2,3}}}
             {{1,1},{2,3}}    {{{1,1}},{{2},{3}}}
             {{1,2},{1,2}}    {{{1}},{{2},{1,1}}}
             {{1,2},{1,3}}    {{{1,2}},{{1},{1}}}
             {{1},{2,3,4}}    {{{1}},{{2},{1,2}}}
             {{1,2},{3,4}}    {{{1,2}},{{1},{2}}}
             {{2},{1,1,1}}    {{{1}},{{2},{1,3}}}
             {{2},{1,1,3}}    {{{1,2}},{{1},{3}}}
             {{1},{1},{1,1}}  {{{1}},{{2},{3,4}}}
             {{1},{1},{1,2}}  {{{1,2}},{{3},{4}}}
             {{1},{1},{2,2}}  {{{2}},{{1},{1,1}}}
             {{1},{1},{2,3}}  {{{2}},{{1},{1,3}}}
             {{1},{2},{1,1}}  {{{2}},{{3},{1,1}}}
             {{1},{2},{1,2}}  {{{2,3}},{{1},{1}}}
             {{1},{2},{1,3}}
             {{1},{2},{3,4}}
             {{2},{3},{1,1}}

Crossrefs

Labeled versions are A330475 (strongly normal) and A330655 (normal).

The case where the atoms are all different is A318813.

The case where the atoms are all equal is (also) A318813.

The labeled case of set partitions is A005121.

The labeled case of integer partitions is A330679.

The case of maximal depth is A330663.

The version where leaves are sets (as opposed to multisets) is A330668.

Cf. A000311, A000669, A001678, A002846, A004114, A007716, A048816, A213427, A306186, A320154, A320160, A330470, A330666.

Sequence in context: A341214 A106159 A160915 * A277501 A277503 A381382

Adjacent sequences: A330471 A330472 A330473 * A330475 A330476 A330477

Keyword

nonn,more

Author

Gus Wiseman, Dec 26 2019

Status

approved