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