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According to this MO answer Koszul duality is related to operations on generating series;

1) multiplicative inversion for quadratic algebras,

2) compositional inversion for quadratic operads,

3) Legendre transformation (disguised comp. inversion) for cyclic quadratic operads.

Chapoton, Vallette, Loday, and others have used binary trees to characterize these relationships. There are numerous combinatoric structures related to these operations, including permutohedra and mappings of weighted surjections for forming the reciprocal of exponential generating series, and Stasheff polytopes (type A associahedra) for compositional inversion of ordinary generating series.

What combinatoric/geometric structures, do you feel, give you the most enlightening insights on the relationships between these inversions and Koszul duality? (with some comment on how/why)

Related MO-Q: sym. polynomials, stirling number reciprocity, and gravity operads.

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  • $\begingroup$ Perhaps item 2 is addressed by Drakes' thesis "An inversion theorem for labelled trees ..." people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf. $\endgroup$ Dec 8, 2015 at 13:59
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    $\begingroup$ Related: "A Quillen adjunction between algebras and operads, Koszul duality, and the Lagrange inversion formula" by Dotsenko arxiv.org/abs/1606.08222 $\endgroup$ Jan 12, 2017 at 20:50
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    $\begingroup$ See also the youtube video Graded Algebras and the Lagrange Inversion Formula by Dotsenko. $\endgroup$ Jan 23, 2017 at 13:09
  • $\begingroup$ Cf. "Trialgebras and families of polytopes" by Loday and Ronco arxiv.org/abs/math/0205043 $\endgroup$ Mar 4, 2017 at 22:35
  • $\begingroup$ Cf. math.ucr.edu/home/baez/week238.html, John Baez's discussion of the relations among the Maurer-Cartan form, Lie differential forms, the Jacobi identity, and Koszul duality for the associative, commutative, and Lie algebras. $\endgroup$ Jul 8, 2018 at 1:05

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