2025/06/17 by Alex Aguila, Aguila, Alex, Elvis Cabrera +3
Computer Science · Mathematics · #06A07 #Advanced Algebra and Logic #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2506.14892
openalex publication_date 2025/06/17 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
This article investigates atomic decompositions in geometric lattices isomorphic to the partition lattice Π(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 \mathfrakN\colon Π(X) → ℕ, which assigns to each partition π the number of minimal atomic decompositions of π. We then consider a distinguished subset of atoms, R, referred to as the set of red atoms, and derive a recursive formula for \pmbπ(X, j, s, R), which enumerates the rank-j partitions expressible as the join of exactly s red atoms.