In the book "Series Expansion Methods" by Oitmaa, Hamer, and Zheng, Appendix 6, they define a moment $\left<\,\,\right>$ as the average of a set of variables, and then define the cumulant $\left[\,\,\,\right]$ as follows
\begin{equation}\left<\alpha\cdots\zeta\right>=\sum_{P}\left[\alpha\cdots\beta\right]\left[\gamma\cdots\zeta\right]\tag{1} \end{equation}
where the sum goes over all possible partitions $P$ of the set of variables, or symbolically
$$\left<\,\,\right> = \sum_{n=1}^\infty \frac{1}{n!}\left[\,\,\,\right]^n\tag{2}$$
For example,
\begin{align*} \left<\alpha\right>&=[\alpha]\\ \left<\alpha\beta\right>&=[\alpha\beta]+[\alpha][\beta] \\ \left<\alpha\beta\gamma\right>&=[\alpha\beta\gamma]+[\alpha\beta][\gamma]+[\beta\gamma][\alpha] + [\gamma\alpha][\beta]+[\alpha][\beta][\gamma] \tag{3} \end{align*}
[which] can be inverted to give symbolically
$$\left[\,\,\,\right] = \sum_{n=1}^\infty (-1)^{n-1}\frac{1}{n}\left<\,\,\right>^n\tag{4}$$ for example \begin{align*} [\alpha]&=\left<\alpha\right>\\ [\alpha\beta]&=\left<\alpha\beta\right>-\left<\alpha\right>\left<\beta\right>\\ [\alpha\beta\gamma]&=\left<\alpha\beta\gamma\right>-\left<\alpha\beta\right>\left<\gamma\right>-\left<\beta\gamma\right>\left<\alpha\right> - \left<\gamma\alpha\right>\left<\beta\right>+2\left<\alpha\right>\left<\beta\right>\left<\gamma\right>\tag{5} \end{align*}
These "symbolic" equations are quite mystifying to me. I see how to make the inversion by identifying the first as $e^{[\,]}-1$ and obtaining the second by expanding $\ln(1+\left<\,\,\right>)$, but I don't understand how to interpret either of these equations to write down the corresponding expressions as given, since it is not clear to me what $[\,\,\,]^n$ and $\left<\,\,\right>^n$ mean.
I would like to understand
(a) what is the relationship between equations (1) and (2),
(b) how "rigorous" is the relationship between equations (2) and (4), and
(c) how am I to interpret equation (4) to obtain expressions in (5) (and the same for (2) and (3))
Note I'm a physicist (not a mathematician per se), and this seems incredibly hand-waving to me, but also rather elegant. I have seen more "proper" definitions of cumulants in terms of generating functions.