Characterize primes in the ring of partitions

Characterize the prime elements (i.e., multiplicatively irreducible elements) in the ring of partitions P, where an element is an integer partition n=(n1,...,nk), addition is defined by concatenation of parts, and multiplication is defined by forming the partition with parts n_i m_j for all i and j. Determine a complete criterion that identifies all such primes, beyond the known sufficient cases where either the sum |n|=∑_j n_j is a rational prime or the length k is a rational prime.

Background

The paper defines a ring structure on integer partitions: addition is concatenation and multiplication is obtained by taking all pairwise products of parts, aligning with the structure of multipartite graphs under Zykov join and Sabidussi product. In this setting, a “partition prime” refers to an element that cannot be factored nontrivially under the ring multiplication.

Known sufficient conditions for multiplicative primality are provided: a partition n=(n1,...,nk) is prime if either its total |n|=∑_j n_j is a rational prime or its length k is a rational prime. However, examples such as (3,4,5,2) show that primes exist even when |n| and k are both composite, motivating a complete characterization.