  A frame in an $n$-dimensional Hilbert space $H_n$ is a possibly redundant
collection of vectors $\{f_i\}_{i\in I}$ that span the space. A tight frame is
a generalization of an orthonormal basis. A frame $\{f_i\}_{i\in I}$ is said to
be scalable if there exist nonnegative scalars $\{c_i\}_{i\in I}$ such that
$\{c_if_i\}_{i\in I}$ is a tight frame. In this paper we study the
combinatorial structure of frames and their decomposition into tight or
scalable subsets by using partially-ordered sets (posets). We define the factor
poset of a frame $\{f_i\}_{i\in I}$ to be a collection of subsets of $I$
ordered by inclusion so that nonempty $J\subseteq I$ is in the factor poset if
and only if $\{f_j\}_{j\in J}$ is a tight frame for $H_n$. A similar definition
is given for the scalability poset of a frame. We prove conditions which factor
posets satisfy and use these to study the inverse factor poset problem, which
inquires when there exists a frame whose factor poset is some given poset $P$.
We determine a necessary condition for solving the inverse factor poset problem
in $H_n$ which is also sufficient for $H_2$. We describe how factor poset
structure of frames is preserved under orthogonal projections. We also consider
the enumeration of the number of possible factor posets and bounds on the size
of factors posets. We then turn our attention to scalable frames and present
partial results regarding when a frame can be scaled to have a given factor
poset.
