I deal with a lot of multi-sigma functions, which I assign compact formulas through a step-wise "pull-back". It is best explained through a well-known example:
$$\begin{align} \sum_{i_1=1}^n \sum_{i_2=1}^{i_1}\cdots \sum_{i_k=1}^{i_{k-1}} 1 &= \sum_{i_1=1}^n \sum_{i_2=1}^{i_1}\cdots \sum_{i_{k-1}=1}^{i_{k-2}} i_{k-1} \\[2ex]&= \sum_{i_1=1}^n \sum_{i_2=1}^{i_1}\cdots \sum_{i_{k-2}=1}^{i_{k-3}}\binom{i_{k-2}+1}2 \\[2ex]& \ \ \vdots \\[2ex] &= \binom{n+ k-1}k \end{align} $$
What kind of terminology/notation may I use to refer to the "pull-backs" at each step? I wonder, because I want to talk about things like the formula for how $f(i_j)$ changes into $f(i_{j-1})$ with each "pull-back". Any pointers to literature dealing with these kinds of topics would be greatly appreciated, as I'd like a more general comfort in speaking about this kind of symbolic manipulation. But more specifically, my question is, how can I talk about these "pull-backs"?