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Ways of partitioning an n-multiset with multiplicities some partition of n.

Number of multiset partitions of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities. The (weakly) normal version is A255906. - Gus Wiseman, Dec 31 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. For an integer partition the T_0 condition means the dual of the multiset partition obtained by factoring each part into prime numbers is strict (no repeated blocks).

Also the number of integer partitions of n with no equivalent primes. In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent. See A316978 for more examples.

Consider the "ordered partitions of partitions" as described in A055887. They are produced by introducing separators (a term used by J. Riordan) between the parts of a partition. If a partition has P parts, then it is possible to introduce 1, 2, ... P-1 separators. Let "|" denote such a separator. We just append 1,2,...,P-1 separators to each integer partition of n and subsequently form all permutation of the resulting list (which is composed of parts and separators).

There are some rules: If we do not append a separator, then we do not perform any permutation. Furthermore, we do not accept permutations which have a dangling separator in front of the integer parts or past them. E.g. the permutations [|,1,2,3] and [1,2,3,|] are forbidden. Furthermore, sequences of separators as "|,|" are forbidden.

Now we impose a further restriction on the permutations. Consider the elements between two separators. We call their number "occupation number". We just request that the occupation number of a ordered partition is monotonically decreasing (if we start from the left to the right of a permutation written in our notation). If we interpret a separator as a level, then we can speak of a hierarchy. E.g. we do not count [1,|,2,3,|,4] as a hierarchy, but we accept [1,2|,3,4] as a hierarchy. We thus speak of "hierarchically ordered partitions of partitions" for this sequence.

With the generating function f := z -> 1/(mul(1-z^i/mul(1-z^j,j=1..i), i=1..25)); we get the asymptotic expansion using the command equivalent (f(z),z,n);

The result is 3.788561346*exp(-n)^(-log(2)) + O(1/n*exp(-n)^(-log(2))). Let fas := n -> 3.788562346*exp(-n)^(-log(2)); then for n=60 we get fas(60)/A141199(60)= .4367915009e19/4344507472742893655 = 1.005387846.

In short, a(n) is the number of finite sequences of integer partitions with weakly decreasing lengths and total sum n. The case of twice-partitions is A358831. A version choosing compositions is A218482. The strictly decreasing case is A358836. For ordered set partitions we have A005651. For weakly decreasing bigomega see A358335. - Gus Wiseman, Dec 05 2022

Also the number of integer compositions of n whose leaders of maximal weakly decreasing runs are strictly increasing. For example, the composition (1,2,2,1,3,1,4,1) has maximal weakly decreasing runs ((1),(2,2,1),(3,1),(4,1)), with leaders (1,2,3,4), so is counted under a(15). - Gus Wiseman, Aug 21 2024

If these partitions are "flattened" into a simple partition, the resulting partitions are those for which any part size present with multiplicity k implies the presence of at least k(k-1)/2 larger parts. E.g., [3,1|1] flattens to [3,1^2], 1 has multiplicity 2, so there must be at least 2*1/2 = 1 part larger than 1 - which is the 3.

Matches A072706 for n < 10, since a unimodal composition into distinct parts can be placed uniquely as a hook. Starting with n = 10, additional partitions are possible (starting with [4,3|2,1] and [4,2|3,1]).

a(n) is the number of finite sets of compositions with total sum n. The case of constant sums is A358904, cf. A074854. The case of distinct sums is A304961, ordered A336127. The ordered version (sequences of distinct compositions) is A358907. - Gus Wiseman, Dec 12 2022