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 $\mathrm{\Delta }$ on a vertex set $V$, together with an ordered partition $(V_1,\dots ,V_n)$ of $V$, such that every face $F$ of $\mathrm{\Delta }$ satisfies $\mathrm{\mid }F\cap V_i\mathrm{\mid }\le a_i$. For each $$\mathbf{b}$ = (b_1, \dots, b_n) \leq $\mathbf{a}$$,let$f_{$\mathbf{b}$}$bethenumberoffaces$F$of${\Delta}$ such that $\mathrm{\mid }F\cap V_i\mathrm{\mid }=b_i$. The array of integers $\{f_{\mathbf{b}}{\}}_{\mathbf{b}\le \mathbf{a}}$ is called the fine $f$-vector of $\mathrm{\Delta }$, and it is a refinement of the $f$-vector of $\mathrm{\Delta }$. 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 $\mathrm{\Delta }$ 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.