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Thinking of some questions of homotopical algebra for operads, I ended up with a following question, perhaps someone will recognize something here:

Let $\{a_n\}_{n\ge 2}$ be a sequence of nonnegative integers. Let's say that this sequence has the property S if for the series $$ f(t)=t+\sum_{n\ge 2} \frac{a_n}{n!}t^n, $$ the series $$ f^{<-1>}(f(t)-t) $$ has non-negative coefficients. Here $f^{<-1>}$ is the reversion (compositional inverse) of $f$. For example, the sequence $a_n=(n-1)!$ has the property S and the sequence $a_n=n!$ does not, these are easy exercises. It seems more difficult (but true) that the sequence $a_n=n^{n-1}$ has the property S.

My question is whether someone saw something of that sort before, or has some relevant references. In particular, I am interested to know if can prove in a very indirect way that the sequence $\{a_n\}$ for which $f^{<-1>}(t)=t-t^2+\frac{t^5}{120}$ has the property S, but I am wondering if the same can be established by a direct combinatorial argument of some sort.

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  • $\begingroup$ Related: mathoverflow.net/questions/291888/… $\endgroup$ Jul 9, 2022 at 17:28
  • $\begingroup$ @TomCopeland can you perhaps clarify? Do you mean "related" in some vague way or can you indicate some precise mathematical relationship (which is not at all obvious to me)? $\endgroup$ Jul 11, 2022 at 6:49
  • $\begingroup$ "My question is whether someone has seen something of that sort before?" It's of the form of a specialization of a formal group law of the type Abel addressed $FGL(x,y)= f^{(-1)}(f(x)+f(y))$ and of the type Charles Graves investigated $\exp[ug(x)\partial_x]W(x) = W[f^{(-1)}(f(x)+u)]$ in the 1800s, where $g(x) = 1/\partial_xf(x)$. $\endgroup$ Jul 11, 2022 at 14:41
  • $\begingroup$ Jair Taylor, who authored the MO-Q I alluded to, was interested in combinatorial interpretations and positivity of such FGLs. $\endgroup$ Jul 11, 2022 at 14:47
  • $\begingroup$ @TomCopeland that I see but I do not see at all how the properties "$f^{<-1>}(f(x)+f(y))$ has non-negative coefficients" and "$f^{<-1>}(f(t)-t)$ has non-negative coefficients" are related in any precise way, hence my comment. $\endgroup$ Jul 12, 2022 at 14:30

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