Factorizations in Geometric Lattices

This article investigates atomic decompositions in geometric lattices isomorphic to the partition lattice $\Pi(X)$ of a finite set $X$, a fundamental structure in lattice theory and combinatorics. We explore the role of atomicity in these lattices, building on concepts introduced by D.D. Anderson, D.F. Anderson, and M. Zafrullah within the context of factorization theory in commutative algebra. As part of the study, we first examine the main characteristics of the function $\mathfrak{N}\colon \Pi(X) \rightarrow \mathbb{N}$, which assigns to each partition $\pi$ the number of minimal atomic decompositions of $\pi$. We then consider a distinguished subset of atoms, $\mathcal{R}$, referred to as the set of red atoms, and derive a recursive formula for $\pmb{\pi}(X, j, s, \mathcal{R})$, which enumerates the rank-$j$ partitions expressible as the join of exactly $s$ red atoms.