Is there a name for $2^n$ in combinatorics? $2^n$, with $n$ corresponding to the number of elements, is an almost unavoidable calculation involved in counting. At the root of its usefulness there lies - I presume - the identity:
$$\sum_{0\leq{k}\leq{n}}\binom nk =2^n$$
that explains its application to the calculation of the power set or $2^S$ or the set of all subsets of $S$. However, it is also used without much need for mathematical theory in day to day calculations, such as determining the different ways that $+$'es and $-$'es can be assigned to a list of numbers, say from $1$ to $3$ (i.e. $n=3$): $1$ can be positive  or negative ($2$), and for each branch, $2$ can be positive or negative ($2^2)$, and for each choice, $3$, in turn, can assume positive or negative signs, for a total of $2^3$ ways of allocating signs.  
In looking up for a name akin to combinations for $\binom nk$, or permutations for $P(n,k)$ for this ubiquitous $2^n$ operation, I have found "number of $k$-combinations for all $k$, with an explanation as to its correspondence to the sum of the $n$-th row of Pascal triangle.
Very neat. But is there a mathematical name that directly and succinctly identifies this $2^n$ operation of counting a process for which each node or step splits into a binary choice?

I don't think there is a short name for $2^n$. However, I think that "the number of subsets of a set with $n$ elements" (or cardinal of the powerset, of course) is already nice enough. For instance, it fits very well with your exemple of +'es and -'es: just select the subset of positive terms.
