OFFSET
0,4
COMMENTS
We define a multiset partition to be balanced if it has exactly as many parts as the greatest size of a part.
EXAMPLE
Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions (empty column indicated by dot):
{{1}} . {{1},{1,1}} {{1,1},{1,1}} {{1},{1},{1,1,1}}
{{1},{2,2}} {{1,1},{2,2}} {{1},{1},{1,2,2}}
{{1},{2,3}} {{1,2},{1,2}} {{1},{1},{2,2,2}}
{{2},{1,2}} {{1,2},{2,2}} {{1},{1},{2,3,3}}
{{1,2},{3,3}} {{1},{1},{2,3,4}}
{{1,2},{3,4}} {{1},{2},{1,2,2}}
{{1,3},{2,3}} {{1},{2},{2,2,2}}
{{1},{2},{2,3,3}}
{{1},{2},{3,3,3}}
{{1},{2},{3,4,4}}
{{1},{2},{3,4,5}}
{{1},{3},{2,3,3}}
{{1},{4},{2,3,4}}
{{2},{2},{1,2,2}}
{{2},{3},{1,2,3}}
{{3},{3},{1,2,3}}
PROG
(PARI) \\ See A340652 for G.
seq(n)={Vec(1 + sum(k=1, n, polcoef(G(n, n, k, y), k, y) - polcoef(G(n, n, k-1, y), k, y)))} \\ Andrew Howroyd, Jan 15 2024
CROSSREFS
The version for partitions is A047993.
The co-balanced version is A319616.
The cross-balanced version is A340651.
The twice-balanced version is A340652.
The version for factorizations is A340653.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A098124 counts balanced compositions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340597 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 05 2021
EXTENSIONS
a(11) onwards from Andrew Howroyd, Jan 15 2024
STATUS
approved