2013/06/07 by Kai Fong Ernest Chong, Chong, Kai Fong Ernest
Mathematics · #05E45 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1306.1787
openalex publication_date 2013/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A colored complex of type a = (a1, …, an) is a simplicial complex Δ on a vertex set V, together with an ordered partition (V1, …, Vn) of V, such that every face F of Δ satisfies |F ∩ Vi| ≤ ai. For each b = (b1, …, bn) ≤ a, let fb be the number of faces F of Δ such that |F ∩ Vi| = bi. The array of integers \fb\b ≤ a is called the fine f-vector of Δ, and it is a refinement of the f-vector of Δ. In this paper, we generalize the notion of Macaulay representations and give a numerical characterization of the fine f-vectors of colored complexes of arbitrary type, in terms of these generalized Macaulay representations. As part of the proof, we introduce the property of a-Macaulay decomposability for simplicial complexes, which implies vertex-decomposability, and we show that every pure color-shifted balanced complex Δ of type a is a-Macaulay decomposable. Combined with previously known results, we also obtain a numerical characterization of the flag f-vectors of completely balanced Cohen-Macaulay complexes.