Algebraic structures in integer partitions

Abstract

We study various algebraic structures based on the derived partitions of a partition λ of a positive integer n which are defined as partitions π of positive integers m ≤ n whose parts form a sub-multiset of the multiset of the parts of λ. We show that the set of derived partitions forms a group under addition and a monoid under multiplication. A ring isomorphism is established with the direct product of the rings of residue classes modulo a positive integer, where the number factors of the product is equal to the number of distinct parts of λ. Using the algebraic tool provided by the ring of derived partitions we explore new properties of perfect partitions. We extend the latter to pseudo perfect partitions, that is, partitions whose parts contain at most one partition of a smaller weight. We also classify the types of additive cyclic groups of derived partitions and give a rule for the determination of subgroups. As applications, we study k-complete partitions of n whose parts contain all the partitions of 1 to n. We found some properties of the class of k complete partitions with least weight, and the class of strictly k-complete partitions, using generating functions and combinatorial methods. We also give a brief discussion of perfect-partition-complete partitions which are single partitions that contain all the perfect partitions of a given number.