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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. $\endgroup$ – Tom Copeland Dec 8 '15 at 13:59
  • $\begingroup$ Related: "A Quillen adjunction between algebras and operads, Koszul duality, and the Lagrange inversion formula" by Dotsenko arxiv.org/abs/1606.08222 $\endgroup$ – Tom Copeland Jan 12 '17 at 20:50
  • $\begingroup$ See also the youtube video Graded Algebras and the Lagrange Inversion Formula by Dotsenko. $\endgroup$ – Tom Copeland Jan 23 '17 at 13:09
  • $\begingroup$ Cf. "Trialgebras and families of polytopes" by Loday and Ronco arxiv.org/abs/math/0205043 $\endgroup$ – Tom Copeland Mar 4 '17 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. $\endgroup$ – Tom Copeland Jul 8 '18 at 1:05

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