Derivatives of exp(f(x)) and partitions of an set

I am trying to express the $k$-th derivative of $g(x) = exp(f(x))$ in a meaningful way. My logic got me so far. For the first derivative, the result would be $$g' = f' g$$ Second derivative would be $$g'' = (f'' + f'^2) g$$ Third derivative is $$g''' = f''' + 3 f'' f' + f'^3$$ and so on.

The correspondence that I made was that each derivative can be expressed as $$f ''' \equiv (0,0,3)$$ $$f'f'' = (0,1,2)$$ and $$f'^3 \equiv (1, 1, 1).$$ This is identical to representing the partitions of the integer 3. This is similar to the partitions of a set; a set of size 3, $\left\{a,b,c\right\}$, can be partitioned into $$\left\{a,b,c\right\} \equiv(0,0,3)$$ $$\left\{a\right\} \left\{b,c\right\}, \left\{b\right\} \left\{a,c\right\}, \left\{c\right\} \left\{a,b\right\} \equiv(0,1,2)$$ $$\left\{a\right\} \left\{b\right\} \left\{c\right\} \equiv(1,1,1).$$ The coefficient, for example, of $f'f''$, is the number of partitions corresponding to $(0,1,2)$. Then, I know that these coefficients can be calculated with the help of multinomial coefficients.

My question is: is there a proof of (or a reference about) this correspondence between the derivatives of $g$ and counting the number of partitions?

• Have you tried induction? Seems like a reasonable idea here. – ajotatxe Feb 6 '17 at 19:13
• – Phicar Feb 6 '17 at 19:17
• I'm not sure, but this might relate to the Butcher group: en.wikipedia.org/wiki/Butcher_group – Mosquite Feb 7 '17 at 0:34

It derives the $n$th derivative of composite functions by means of partitions resp. compositions, their ordered companions.
Note: If we consider an integer $n$ as (ordered) sum with $k$ parts $$n=\pi_1+2\pi_2+\cdots +k\pi_k$$ we can represent it as $k$-tuple \begin{align*} (\pi_1,\pi_2,\ldots,\pi_k) \end{align*} Similarly we can represent \begin{align*} f^{\prime\prime\prime} &\equiv (0,0,1)\\ f^{\prime}\cdot f^{\prime\prime}&\equiv (1,1,0)\\ \left(f^{\prime}\right)^3&\equiv (3,0,0) \end{align*}