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

0,3

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.

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

Sequence in context: A148127 A148128 A148129 * A123432 A345216 A151398

Adjacent sequences: A340648 A340649 A340650 * A340652 A340653 A340654

Keyword

nonn

Author

Gus Wiseman, Feb 05 2021

Extensions

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

Status

approved