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.
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
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.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 26 2019
STATUS
approved