We study strongly irreducible partitions, namely integer partitions whose parts are pairwise coprime across different positions. The aim of the paper is to replace an informal proposal with a logically precise theorem-based article. We give stable definitions, isolate the role of the part 1, introduce the ones-free core, and prove structural theorems on repeated parts, prime-support separation, length bounds, and decomposition identities. On the counting side, we derive a general fixed-length reduction to ones-free layers and use it to organize the first three counting blocks. The first block gives exact formulas for the one-part and two-part cases, the three-part reduction, the stable near-maximal length regime, and global lower and upper bounds for the total counting function. The second and third blocks treat the ones-free three-part and four-part layers through exact parametrizations, admissible-region reformulations, Möbius-weighted formulas, explicit initial values, nonvanishing families, and theorem-level upper bounds. We also prove a generating-series factorization induced by the ones-free core, formulate a logically exact state-space recursion, and develop an associated graded-algebra framework for the ones-free layers through part-marked series, quotient presentations, tensor decompositions over disjoint prime blocks, and restricted Hilbert-series factorizations. More speculative directions such as normalization procedures are recorded only as open problems. In this way, the paper provides a theorem-dense and logically safe starting point for a systematic theory of strongly irreducible partitions.
Jianming Wang (Thu,) studied this question.