Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.
by Gus Wiseman, Jul 12 2019

Notes:
1. A set partition is a finite set of disjoint finite sets.
2. A factorization is a multiset of positive integers > 1.
3. A multiset partition is a finite multiset of finite nonempty multisets. It is normal if it covers an initial interval of positive integers.
4. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.


Set partitions

- Lengths equal: A038041
- Lengths different: A007837
- Sums equal: A035470
- Sums different: A275780
- Averages equal: A326512
- Averages different: A326513


Factorizations

- Lengths equal: A322794
- Lengths different: A326514
- Sums equal: A321455
- Sums different: A321469
- Averages equal: A326515
- Averages different: A326516


Normal multiset partitions

- Lengths equal: A317583
- Lengths different: A326517
- Sums equal: A326518
- Sums different: A326519
- Averages equal: A326520
- Averages different: A326521


MM-numbers

- Lengths equal: A320324
- Lengths different: A326533
- Sums equal: A326534
- Sums different: A326535
- Averages equal: A326536
- Averages different: A326537