Characterize primes in the ring of partitions

Characterize the prime elements (i.e., multiplicatively irreducible partitions) in the ring of integer partitions whose elements are partitions n=(n1,...,nk), with addition defined by concatenation n+m=(n1,...,nk,m1,...,ml) and multiplication defined by forming all pairwise products n*m=(n_i m_j) for 1≤i≤k and 1≤j≤l. Determine precisely which partitions are multiplicatively prime beyond the known sufficient cases where either the sum of parts |n| is a rational prime or the number of parts k is a rational prime.

Background

The paper introduces the ring of partitions as a commutative ring where addition is concatenation of parts and multiplication is given by distributing pairwise products of parts. It proves that the maps n↦|n| (sum of parts) and n↦k(n) (number of parts) are ring homomorphisms to the integers, implying that partitions with |n| prime or k(n) prime are multiplicative partition primes.

However, the author notes the existence of multiplicative primes not covered by these simple criteria, such as p=(3,4,5,2), which cannot be factored into nontrivial partition factors despite |p|=14 and k(p)=4 being composite. This motivates a general characterization of all prime elements in the ring of partitions.