Easy way of seeing if swapping summation is ok? (Generating functional derivation of Bell numbers)

Question

On page 21 of his book generatingfunctionology (available for free on the author's homepage), the author rearranges the summations in the following way:

\begin{align} b(n) &= \sum^M_{k=1} \sum^k_{r=1} (-1)^{k-r} \frac{r^{n-1}}{(r-1)!(k-r)!} \\ &= \sum^M_{r=1} \frac{r^{n-1}}{(r-1)!} \sum^M_{k=r} \frac{(-1)^{k-r}}{(k-r)!} \end{align}

It is not immediately obvious to me that this is in fact correct. I have convinced myself for some values of $M$ that I indeed get the same set of ordered pairs $(k,r)$. But is there a way that I can immediately show that the second line is equivalent to the first?

Answer 1

Hint: Maybe the following representation of indices is helpful

\begin{align*} \sum_{k=1}^{M}\sum_{r=1}^{k}b_{r,k}=\sum_{1\leq r\leq k\leq M}b_{r,k}=\sum_{r=1}^{M}\sum_{k=r}^Mb_{r,k} \end{align*}