Generalized Macaulay representations and the flag $f$-vectors of generalized colored complexes

A colored complex of type $\mathbf{a} = (a_1, \dots, a_n)$ is a simplicial complex ${\Delta}$ on a vertex set $V$, together with an ordered partition $(V_1, \dots, V_n)$ of $V$, such that every face $F$ of ${\Delta}$ satisfies $|F \cap V_i| \leq a_i$. For each $\mathbf{b} = (b_1, \dots, b_n) \leq \mathbf{a}$, let $f_{\mathbf{b}}$ be the number of faces $F$ of ${\Delta}$ such that $|F \cap V_i| = b_i$. The array of integers $\{f_{\mathbf{b}}\}_{\mathbf{b} \leq \mathbf{a}}$ is called the fine $f$-vector of ${\Delta}$, and it is a refinement of the $f$-vector of ${\Delta}$. 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 $\mathbf{a}$-Macaulay decomposability for simplicial complexes, which implies vertex-decomposability, and we show that every pure color-shifted balanced complex ${\Delta}$ of type $\mathbf{a}$ is $\mathbf{a}$-Macaulay decomposable. Combined with previously known results, we also obtain a numerical characterization of the flag $f$-vectors of completely balanced Cohen-Macaulay complexes.