The *formal group law* associated with a generating function $f(x) = x + \sum\_{n=2}^\infty a\_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my [thesis](https://digital.lib.washington.edu/researchworks/handle/1773/36757), I found a large number of examples of formal group laws that have combinatorial interpretations and thus have nonnegative coefficients. In Sec 9.1 I conjectured the following characterization for positivity of a formal group law:


**Conjecture.** $f(f^{-1}(x) + f^{-1}(y))$ has nonnegative coefficients if and only if $$\phi(x) = \frac{1}{\frac{d}{dx} f^{-1}(x)}$$ has nonnegative coefficients.


At least one direction is easy: The positivity of the FGL implies positivity of $\phi(x)$.


I have not been able to prove the converse, but there is some evidence. Start with $$\phi(x) = 1 + t\_1x + t\_2\frac{x^2}{2!} + t\_3\frac{x^3}{3!} + \cdots$$ for indeterminates $t\_i$ and define $f(x)$ by $f(0) = 0$, $1/(f^{-1})'(x) = \phi(x)$, or equivalently, $f'(x) = f(\phi(x))$. Then we can compute the coefficients of $f(f^{-1}(x) + f^{-1}(y))$ and they seem to all be polynomials with nonnegative coefficients in the variables $t\_i$. 


Often it is more illuminating to consider the slightly more general symmetric function $$F = f(f^{-1}(x\_1) + f^{-1}(x\_2) + \cdots).$$ The expansion of $F$ in the monomial basis of the ring of symmetric functions is


\begin{align\*}
F = m\_1 &+ (2t\_1)\frac{m\_{11}}{2!} + (3t\_2)\frac{m\_{21}}{3!} + (6t\_1^2 + 6t\_2)\frac{m\_{111}}{3!}\\
 &+ (4t\_3)\frac{m\_{31}}{4!} + (12t\_1t\_2 + 6t\_3)\frac{m\_{22}}{4!} + (36t\_1t\_2 + 12t\_3)\frac{m\_{211}}{4!} \\
&+ (24t\_1^3 + 96t\_1t\_2 + 24t\_3)\frac{m\_{1111}}{4!} + (5t\_4)\frac{m\_{41}}{5!} + (30t\_2^2 + 30t\_1t\_3 + 10t\_4)\frac{m\_{32}}{5!}\\ &+ (60t\_2^2 + 80t\_1t\_3 + 20t\_4)\frac{m\_{311}}{5!} + (120t\_1^2t\_2 + 120t\_2^2 + 150t\_1t\_3 + 30t\_4)\frac{m\_{221}}{5!}\\
&+ (420t\_1^2t\_2 + 240t\_2^2 + 360t\_1t\_3 + 60t\_4)\frac{m\_{2111}}{5!} \\
&+ (120t\_1^4 + 1320t\_1^2t\_2 + 480t\_2^2 + 840t\_1t\_3 + 120t\_4)\frac{m\_{11111}}{5!}\\
&+ \cdots
\end{align\*}


Note that $f(x)$ here has a combinatorial interpretation due to [Bergeron-Flajolet-Salvy](https://hal.inria.fr/inria-00074977/document): $f(x)$ is the exponential generating function for increasing trees weighted by their degree sequence in the variables $t\_i$. So there is reason to think that there is a combinatorial interpretation of $F$ in terms of increasing trees. 


An interesting special case if $\phi(x) = 1 + x^2$, so that $f(x) = \tan(x)$. Then the associated formal group law is a sum of Schur functions of staircase-ribbon shape: $$f(f^{-1}(x\_1) + f^{-1}(x\_2) + \cdots ) = \sum\_{n=1}^\infty s\_{\delta\_{n} / \delta\_{n-2}}$$ where $\delta\_n$ is the partition $(n,n-1, n-2, \ldots, 1)$. (See [Ardila-Serrano](https://arxiv.org/abs/1108.2219), Prop 3.4.) This can also be interpreted in terms of binary increasing trees.


In many examples given in my thesis I found that there was a combinatorial interpretation of the FGL in terms of chromatic symmetric functions, but I was not able to apply those methods to this more general case.


Edit: Tom Copeland suggested I share some of the Sage code I used to generate these coefficients. [Here](https://cocalc.com/share/b18a3b9c-7deb-44a4-bad6-80cfa356d06e/Formal%20group%20law%20positivity.ipynb?viewer=share) is a Jupyter notebook in CoCalc that shows the computations.



Characterizing positivity of formal group laws
Given $\phi(x)\in\mathbb{R}[[x]]$, with $\phi(0)=1$, we have defined $g(x):=\int^x\_0{dt\over \phi(t)}$, $f:=g^{-1}$ and $$F(x,y)=f\big(g(x)+g(y)\big)=\sum\_{n=0}^\infty \psi\_n(x) {y^n\over n!}\in\mathbb{R}[[x,y]].$$ Let's write a recursion for the coefficient sequence $\psi\_n=\partial\_y^nF(x,0)\in\mathbb{R}[[x]]$, solving by series the differential equation satisfied by $F$,


$$\cases{\phi(x)\, F\_x(x,y)=\phi(F(x,y))\\ F(x,0)=x\ .}$$


One finds $\psi\_0=x$, $\psi\_1=\phi,\dots$ . Let's take $\partial\_y^n$ at $ {y=0}$ on both sides. Faà di Bruno:
$$\partial\_y^n\big( \phi\circ F\big)\big|\_{y=0}=\Big(\sum\_{\alpha\in\operatorname{par}[n]} \phi\_y^{(|\alpha|)} (F)\, \prod\_{s\in\alpha} \partial\_y^{|s|}F \Big) \ \Big|\_{y=0} =\sum\_{\alpha\in\operatorname{par}[n]} \phi^{(|\alpha|)}(x) \prod\_{s\in\alpha} \psi\_{|s|}(x) ,$$
(*Legenda:* The sum is indexed on the set of all partitions of $[n]:=\{1,2,\dots,n\}$, and $|\cdot|$ denotes cardinality. The latter equality comes from $F(x,0)=x$ and $\partial\_y^{j}F(x,y)\big|\_{y=0}=\psi\_j$).
Now we isolate the term $\phi'\psi\_n$, that corresponds to the partition $\alpha$ into a single class, from the terms of the sum indexed on the set of non-trivial partitions, with $|\alpha|>1$, denoted $\operatorname{par}^\*[n]$. Note that each of these terms contains more than one factor $\psi\_j$. 


$$\phi\psi'\_n -\phi' \psi\_n =\sum\_{\alpha\in\operatorname{par}^\*[n]} \phi^{(|\alpha|)} \prod\_{s\in\alpha} \psi\_{|s|} .$$
Multiplying by the integrating factor $\phi^{-2}$ , and since $\psi\_n(0)=0$, for $n>1$ 
$$ \psi\_n(x) =\phi(x)\int\_0^x\big(\!\sum\_{\alpha\in\operatorname{par}^\*[n]} \phi^{(|\alpha|)} \prod\_{s\in\alpha} \psi\_{|s|}\,\big)\phi^{-2}\ dt .$$


It is now clear by complete induction that for any $n\ge1$, $\psi\_n$ is equal to $\phi$ times a series with positive coefficients, proving your conjecture.


