I have been trying to prove an identity using a combinatorial argument or another technique. I need to define the following first:
$$R(n,m) = \{ (r_1,r_2,...,r_n): \sum_{i=1}^n r_i = m \text{ and } r_i \text{ is a positive integer }\}.$$
I believe there is a name for this set. If so let me know. I think of it as the set of ordered partitions of m of size n.
The identity I want to prove is the following. For $m \ge 2$:
$$\sum_{n=1}^m \frac{(-1)^n}{n!}\sum_{r \in R(n,m)}\frac{1}{r_1r_2r_3...r_n}=0$$
The reason I know this identity is true is that the left side is the coefficient of $T^m$ in the power series expansion of $\exp (\log(T+1)) = T+1$ where exp and log are defined by $$\exp(T) = \sum_{n=0}^\infty \frac{T^n}{n!}$$ and $$\log(T+1)=-\sum_{n=1}^\infty(-1)^n\frac{T^n}{n}.$$ I am trying to use the identity to show that exp and log as defined by these power series are in fact inverses. Thanks for any help.