The formal group law associated with a generating function $ f(x) = x + \sum_{n=1}^\infty a_n \frac{x^n}{n!}$ is $ $ f(f^{-1}(x) + f^{-1}(y)).$ $ In my thesis, 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))$ have 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: $ 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, 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.