Strongly Irreducible Partitions: Basic Theory and the First Counting Block

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.