OFFSET
0,3
COMMENTS
We define a multiset partition to be cross-balanced if it uses exactly as many distinct vertices as the greatest size of a part.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
EXAMPLE
Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions:
{{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}
{{1},{1}} {{1},{2,2}} {{1,1},{2,2}}
{{2},{1,2}} {{1,2},{1,2}}
{{1},{1},{1}} {{1,2},{2,2}}
{{1},{2,3,3}}
{{3},{1,2,3}}
{{1},{1},{2,2}}
{{1},{2},{1,2}}
{{1},{2},{2,2}}
{{2},{2},{1,2}}
{{1},{1},{1},{1}}
PROG
(PARI) \\ See A340652 for G.
seq(n)={Vec(1 + sum(k=1, n, G(k, n, k) - G(k-1, n, k) - G(k, n, k-1) + G(k-1, n, k-1)))} \\ Andrew Howroyd, Jan 15 2024
CROSSREFS
The co-balanced version is A319616.
The balanced version is A340600.
The twice-balanced version is A340652.
The version for factorizations is A340654.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A047993 counts balanced partitions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 05 2021
EXTENSIONS
a(11) onwards from Andrew Howroyd, Jan 15 2024
STATUS
approved