cp's OEIS Frontend

A set multipartition is intersecting if no two parts are disjoint. The weight of a set multipartition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

A set system is a finite set of finite nonempty sets. It is intersecting if no two parts are disjoint. The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

A multiset partition is intersecting iff no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict.

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

First differs from A327400 at A327400(24) = 4, a(24) = 3.

From Jianing Song, Jun 09 2025: (Start)

Let n = (p_1)^(e_1) * ... * (p_r)^(e_r), then a(n) is the number of partitions of the multiset formed by e_1 1's, e_2 2's, ..., e_r r's such that each pair of parts is either equal or nonintersecting. Let's call such a partition a (e_1,...,e_r)-partition of {1,2,...,r}.

Note that every (e_1,...,e_r)-partition has a base partition by removing duplicates of parts and elements in each part (e.g., {{1,2,2},{1,2,2},{3,3},{4}} -> {{1,2},{3},{4}}), and the base partition is itself a partition on {1,2,...,r}. Since the number of partitions into identical parts of the multiset formed by e_{i_1} (i_1)'s, ..., e_{i_k} (i_k)'s is d(gcd(e_{i_1},...,e_{i_k})), where d = A000005, the number of (e_1,...,e_r)-partitions having base partition P of {1,2,...,r} is Product_{S in P} d(gcd_{i in S} (e_i)). As a result, the number (e_1,...,e_r)-partitions is Sum_{P is a partition of {1,2,...,r}} Product_{S in P} d(gcd_{i in S} (e_i)).

Examples:

# of e_1-partitions = d(e_1);

# of (e_1,e_2)-partitions = d(gcd(e_1,e_2)) + d(e_1)*d(e_2);

# of (e_1,e_2,e_3)-partitions = d(gcd(e_1,e_2,e_3)) + d(gcd(e_1,e_2))*d(e_3) + d(gcd(e_1,e_3))*d(e_2) + d(gcd(e_2,e_3))*d(e_1) + d(e_1)*d(e_2)*d(e_3);

# of (e_1,e_2,e_3,e_4)-partitions = d(gcd(e_1,e_2,e_3,e_4)) + (d(gcd(e_1,e_2,e_3))*d(e_4) + ...) + (d(gcd(e_1,e_2))*d(gcd(e_3,e_4)) + ...) + (d(gcd(e_1,e_2))*d(e_3)*d(e_4) + ...) + d(e_1)*d(e_2)*d(e_3)*d(e_4).

(End)

A thrice-factorization of n is a choice of a twice-factorization of each factor in a factorization of n. Thrice-factorizations correspond to intervals in the lattice form of the multiorder of integer factorizations.