TITLE: A differential equation governing compositional inversion
QUESTION [7 upvotes]: Looking for references for the following theorem.
Given the formal Taylor series/exponential generating function
$$T(z) = \sum_{n \ge 1} a_n \; \frac{z^n}{n!},$$
for which the indeterminates $a_n$ and the independent variable $z$ may be complex, its formal compositional inverse
$$T^{(-1)}(z) = \sum_{n \ge 1} b_n \; \frac{z^n}{n!}=\sum_{n \ge 1} Prt_n(a_1,a_2,...,a_n) \; \frac{z^n}{n!} $$
satisfies the differential equation
$$-\frac{\partial}{\partial a_{n}}\;T^{(-1)}(z)= \frac{\partial}{\partial z}\;\frac{(T^{(-1)}(z))^{n+1}}{(n+1)!}.$$
Edit: Don't quite follow Pietro's derivation (I already had two other derivations of this theorem), but Pietro's inspires the following third derivation;
With $S(\omega,a_n) = T^{(-1)}(\omega,a_n) = z$ and $\omega = T(z,a_n)$, then $z = S(T(z,a_n),a_n)$,
so, suppressing the $a_n$ in the arguments,
$$\partial_{a_n} z = 0 = \partial_T \;S(T(z))\; \partial_{a_n} \;T(z) + \partial_{a_n} \;S(T(z))$$
$$= \partial_T \;S(T(z))\; \frac{z^n}{n!} + \partial_{a_n} \;S(T(z))$$
$$=\partial_T \;S(T(z))\; \frac{(S(T(z)))^n}{n!} + \partial_{a_n} \;S(T(z))$$
$$=\partial_T \; \frac{(S(T(z)))^{n+1}}{(n+1)!} + \partial_{a_n} \;S(T(z)),$$
implying
$$\partial_{\omega} \; \frac{(S(\omega))^{n+1}}{(n+1)!} + \partial_{a_n} \;S(\omega)=0.$$

Instances I had already found of this integrable hierarchy of diff ids / conservation laws are in the section "5.4 Deformation of flow equations" on p. 57 and in Theorem 5.4 on pp. 66 of "On Emergent Geometry of the Gromov-Witten Theory of Quintic Calabi-Yau Threefold" by Jian Zhou.
One reason I ask for more references is that this diff id implies some interesting relations w.r.t. free probability theory, associahedra, and the inviscid Burgers-Hopf differential equation, so I'm hoping to see some perspectives from others on these and will give due credit on such in any of my postings in the future.

Edit (5/28/22): The derivation is more accurately expressed as
$$\partial_{a_n} \; z=0 =\partial_{a_n} \; S(T(z,a_n),a_n) =\partial_{a_n} \; S(\beta_1,\beta_2)$$
$$= (\partial_{\beta_1} \;  S(\beta_1,\beta_2)) \;\partial_{a_n}\beta_1 +(\partial_{\beta_2} \;  S(\beta_1,\beta_2)) \; \partial_{a_n}\beta_2$$
$$ = (\partial_{\beta_1} \;  S(\beta_1,\beta_2)) \;\frac{z^n}{n!} +\partial_{\beta_2} \;  S(\beta_1,\beta_2) $$
$$ = (\partial_{\beta_1} \;  S(\beta_1,\beta_2)) \;\frac{( S(\beta_1,\beta_2))^n}{n!} +\partial_{\beta_2} \;  S(\beta_1,\beta_2) $$
$$ = \partial_{\beta_1} \;   \;\frac{(S(\beta_1,\beta_2))^{n+1}}{(n+1)!} +\partial_{\beta_2} \;  S(\beta_1,\beta_2) $$
which may be expressed as
$$0 = \partial_{u} \;   \;\frac{(S(u,a_n))^{n+1}}{(n+1)!} +\partial_{a_n} \;  S(u,a_n).$$

REPLY [5 votes]: I do not have a reference, but I’d say it is a plain instance of the Implicit Function Theorem for formal power series. I add the computation below, in case you had a different one.
We choose our index $n\ge1$ and see $T$ as a power series in $z$ and $a:=a_{n}$, that is, as an element of $\mathbb{C}[[z, a]]$. The corresponding compositional inverse of $T(z, a)$, is the formal series $S(z, a)$  well defined as implicit function by
$$z=T(S(z, a), a) $$
(with $a_1\neq0$).
Differentiating w.r.to $a $ and multiplying by $\partial_{1  }S(z,a) $ we get
$$0=\Big[\partial_1 T\big(S(z, a ), a \big)\partial_2S(z,a)+\partial_2 T\big(S(z,a),a\big)\big] \partial_1S(z,a)=$$
$$=\partial_1 T\big(S(z, a ), a \big)\partial_1S(z,a) \partial_2S(z,a) +\partial_2T\big(S(z,a),a\big)  \partial_1S(z,a)=$$
$$= \partial_1\big[T (S(z, a ), a)\big]\partial_2S(z,a) + \frac{S(z,a)^{n}}{n!} \, \partial_1S(z,a)=$$
$$=\partial_2S(z,a) + \partial_1\bigg(\frac{S(z,a)^{n+1}}{(n+1)!}\bigg).$$