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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.

Vladimir Dotsenko
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  • Related: https://mathoverflow.net/questions/291888/characterizing-positivity-of-formal-group-laws – Tom Copeland Jul 09 '22 at 17:28
  • @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)? – Vladimir Dotsenko Jul 11 '22 at 06:49
  • "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 $g(x) = 1/\partial_xf(x)$. – Tom Copeland Jul 11 '22 at 14:41
  • Jair Taylor, who authored the MO-Q I alluded to, was interested in combinatorial interpretations and positivity of such FGLs. – Tom Copeland Jul 11 '22 at 14:47
  • @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. – Vladimir Dotsenko Jul 12 '22 at 14:30
  • If you want precisely only an answer, then don't include an implicit vague question such as "My question is whether someone saw something of that sort before". I could answer precisely and simply "Yes," or I could be a real stickler for precision, rather OCD-ish, and ask "What particular instant or period of time in the past are you referring to by the use of the simple past tense of 'see', i.e., 'saw', rather than the present perfect 'has seen'? " A little ambiguity tolerance is required in collegial discussion. Ditto-ish on "perhaps someone will recognize something here." – Tom Copeland Jul 12 '22 at 19:19
  • In any event, $\exp[-t g(x) \partial_x] x; |_{x=t} = f^{(-1)}(f(t)-t)$ with $g(x) =1/f'(x) = 1/ (1 - 2x + x^4/4!)$ with $(g(x)\partial_x)^n x$ presented in OEIS A145271, and Jair addressed a question of when the coefficients of $x^ny^m$ are positive for the more general expression $f^{(-1)}(f(x)+f(y)) = \exp[f(y) g(x) \partial_x] x $ for general $f(x)$. – Tom Copeland Jul 12 '22 at 19:28
  • @TomCopeland I am open to any kind of remarks, but as far as I am concerned, A is more general than B if B is a particular case of A. Your A is not more general, it is vaguely similar. – Vladimir Dotsenko Jul 13 '22 at 05:24

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