Formal definition of random assignment I am looking for a formal definition of random assignment.
Let $\mathbf{Z}$ be a vector of treatment assignments in which each element is 0 (unit not assigned to treatment) or 1 (unit assigned to treatment).  In a JASA article, Angrist, Imbens, and Rubin (1996, 446-47) say that treatment assignment $Z_i$ is random if $\Pr(\mathbf{Z} = \mathbf{c}) = \Pr(\mathbf{Z} = \mathbf{c'})$ for all $\mathbf{c}$ and $\mathbf{c'}$ such that $\iota^T\mathbf{c} = \iota^T\mathbf{c'}$, where $\iota$ is a column vector with all elements equal to 1.
In words, the definition seems to be this: assignment $Z_i$ is random if any vector of assignments that includes $m$ assignments to treatment is as likely as any other vector that includes $m$ assignments to treatment.  
This definition seems unsatisfactory.  What if I decide a priori that I want to rule out a particular vector of assignments, and choose one of the remaining vectors at random?  This practice would not satisfy the AIR definition, but it would still be random assignment.
Here is an example.  Imagine a binary assignment to treatment for each of two subjects.  Let $\mathbf{Z}$ be the vector of treatment assignments.  Then $\mathbf{Z}$ has four possible values: {0, 0}, {0, 1}, {1, 0}, and {1, 1}.  By the AIR definition, assignment is random only if $\Pr(\mathbf{Z} = \{1, 0\}) = \Pr(\mathbf{Z} = \{0, 1\})$.  But why should this be the definition of random assignment, or even a necessary condition for it?  What if I simply decide that I want to rule out {0, 1} and choose at random from the three remaining vectors?  It seems that this practice is consistent with conventional understanding of random assignment but inconsistent with the AIR definition.
So: is there a formal definition of random assignment that encompasses the idea that the experimenter may rule out some assignment vectors a priori?
 A: While Michael Chernick gave a good answer, I do not think that the people who are involved in treatment effect estimation think in terms of finite populations and randomization-based inference. Economists (Angrist and Imbens are well-known econometricians) usually don't; if the OP comes from the same tradition, that is the central issue of this question.
Economists have a model perspective instead, where there's a conceptual population from which units are taken, and there's some sort of an implicit permutation invariance, or "labels do not matter", assumption being made for these sampled units. It is this permutation invariance that is being characterized and quantified in the definition given by the OP. In finite populations, though, every unit is assumed to be unique, and denying a certain treatment for it in the randomization mechanism would produce an unestimable treatment effect. Switching from the model-based inference to randomization-based inference is very difficult; this may have been done in the cited paper, but not very clearly.
A: This definition of random assignment seems to be assigning with equal probability.  To assign 0 weight any of the possible assignments could create bias and should be considered a nonrandom assignemnt by any definition.  However sampling with unequal nonzero weights can be an acceptable procedure (e.g. sampling randomly proportional to size or stratified random sampling with unequal samples per stratum are survey sampling examples).  They fit into a more general definition of random sampling. If one is estimating a mean a weighted average can be used to get a unbiased estimate of the population mean.  By excluding a possible outcome you change the population and it is not apprpriate to draw an inference to the large population that you chose not to sample from.  Also because potential samples were excluded it is impossible to make a weighted adjustment to guarantee an unbiased estimation for the mean of the unrestricted population.
A: One thing that you'll notice in the AIR paper is that they do not condition on covariates $X$. You can generalize the AIR exposition by doing so. 
Let $X$ be an indicator for whether a subject is male. Also suppose that you want men to be more likely to receive treatment than women. You can have $$
\begin{equation*}\Pr(z = c \mid X=1) = \Pr(z = c^\prime \mid X=1)
\end{equation*}$$ and 
$$\begin{equation*}\Pr(z = c \mid X=0) = \Pr(z = c^\prime \mid X=0),\end{equation*}$$ but $$\begin{equation*}\Pr(z = c \mid X=1) > \Pr(z = c \mid X=0)\end{equation*}$$ and still satisfy random assignment in this context. This would justify stratified sampling, for example.
The difference between this generalization and one in which arbitrary vectors are just excluded is that here each person in a particular strata has the same probability of entering treatment, while your suggestion would target specific people to be less likely to enter treatment. If you could do this in a systematic way based upon observable features of the observations, as in the stratification case, you'll be in the clear, but an unsystematic holding back of some members of the population can bias your results.
