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

0,4

Comments

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

Links

Table of n, a(n) for n=0..30.

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.

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

Sequence in context: A097661 A182561 A246915 * A013625 A182929 A361733

Adjacent sequences: A340597 A340598 A340599 * A340601 A340602 A340603

Keyword

nonn

Author

Gus Wiseman, Feb 05 2021

Extensions

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

Status

approved