A340600 Number of non-isomorphic balanced multiset partitions of weight n.

1, 1, 0, 4, 7, 16, 52, 206, 444, 1624, 5462, 19188, 62890, 215367, 765694, 2854202, 10634247, 39842786, 150669765, 581189458, 2287298588, 9157598354, 37109364812, 151970862472, 629048449881, 2635589433705, 11184718653563, 48064965080106, 208988724514022, 918639253237646, 4079974951494828

Offset: 0

Author

Gus Wiseman, Feb 05 2021

Keywords

• nonn

Comments

We define a multiset partition to be balanced if it has exactly as many parts as the greatest size of a part.

Examples

			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}}
		

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.

Cf. A001055, A006141, A007717, A316983, A320663, A324522, A340654, A340655.

Programs

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

Extensions

a(11) onwards from Andrew Howroyd, Jan 15 2024