A340651 Number of non-isomorphic cross-balanced multiset partitions of weight n.

1, 1, 2, 4, 11, 26, 77, 220, 677, 2098, 6756, 22101, 74264, 253684, 883795, 3130432, 11275246, 41240180, 153117873, 576634463, 2201600769, 8517634249, 33378499157, 132438117118, 531873247805, 2161293783123, 8883906870289, 36928576428885, 155196725172548, 659272353608609, 2830200765183775

Offset: 0

Author

Gus Wiseman, Feb 05 2021

Keywords

• nonn

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.

Examples

			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}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

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.

Cf. A001055, A007717, A064174, A316983, A320663, A324522, A339888, A340655.

Programs

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

Extensions

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