Confused by Faà di Bruno's formula notation.

Question

The Faà di Bruno's formula is: $$\frac{d^n}{dt^n}(f(g(x))=\sum\frac{n!}{m_1!1!^{m_1}m_2!2!^{m_2} ... m_n!n!^{m_n}}\cdot f^{(m_1+...+m_n)}(g(x))\cdot \prod_{j = 1}^n \left(g^{(j)}(x) \right)^{m_j}$$ There are three things about this equation that I don't understand. For one the summation sign. There is nothing on it so what do I add up, from where to when? Next where does m come from? what does it stand for? Thirdly is it $\left(m_n!n! \right)^{m_n}$ or $m_n!\left(n! \right)^{m_n}$? Thats a bit unclear considering that the equation was written 1850s and if I remember correctly parenthesis weren't widely used back then. Also can the equation be simplified like this: $$\frac{d^n}{dt^n}(f(g(x))=\sum\frac{n!}{\prod_{i=1}^n m_i!i!^{m_i}}\cdot f^{(\sum_{k=1}^nm_k)}(g(x))\cdot \prod_{j = 1}^n \left(g^{(j)}(x) \right)^{m_j}$$

Answer 1

The sum is over all sets of nonnegative integers $(m_1, m_2,m_3,\dots)$ satisfying the constraint

$${\displaystyle 1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots =n.} $$

This explains also what are $m_i $. For a given $n $ there are only a finite number of such sets (observe that $m_k\le\dfrac nk $ and particularly $m_k=0$ for all $k>n $). This number is known as partition function of $n$.

The correct operation sequence is $$m_k! (k!)^{m_k}. $$