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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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Tom Copeland
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    Perhaps item 2 is addressed by Drakes' thesis "An inversion theorem for labelled trees ..." http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf. – Tom Copeland Dec 08 '15 at 13:59
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    Related: "A Quillen adjunction between algebras and operads, Koszul duality, and the Lagrange inversion formula" by Dotsenko https://arxiv.org/abs/1606.08222 – Tom Copeland Jan 12 '17 at 20:50
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    See also the youtube video Graded Algebras and the Lagrange Inversion Formula by Dotsenko. – Tom Copeland Jan 23 '17 at 13:09
  • Cf. "Trialgebras and families of polytopes" by Loday and Ronco http://arxiv.org/abs/math/0205043 – Tom Copeland Mar 04 '17 at 22:35
  • Cf. http://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. – Tom Copeland Jul 08 '18 at 01:05
  • See Petersen's answer to https://mathoverflow.net/questions/259374/combinatorial-interpretation-of-series-reversion-coefficients – Tom Copeland Sep 05 '18 at 16:46
  • See "The diagonal of the associahedra" by Naruki Masuda, Hugh Thomas, Andy Tonks, Bruno Vallette https://arxiv.org/abs/1902.08059 – Tom Copeland Sep 08 '19 at 13:39
  • See "Monops, monoids and operads: The combinatorics of Sheffer polynomials" by Mendez and Lamoneda (who annoyingly, carefully avoid fully referencing the OEIS) – Tom Copeland Oct 05 '19 at 21:40
  • See related video lecture "Koszul duality in physics" by Dimofte https://lecture2go.uni-hamburg.de/l2go/-/get/v/21960 – Tom Copeland Nov 23 '19 at 22:57
  • "Inversion of integral series enumerating planar trees" by Jean-Louis Loday https://arxiv.org/abs/math/0403316 – Tom Copeland Dec 21 '19 at 23:32
  • "Operads from posets and Koszul duality" by Samuele Giraudo https://arxiv.org/abs/1504.04529 – Tom Copeland Mar 03 '20 at 19:57
  • "Koszul duality for monoids and the operad of enriched rooted trees." by Mendez https://arxiv.org/pdf/0812.4831.pdf – Tom Copeland Aug 25 '21 at 13:46
  • Related: "Moebius specie" by Mendez and Yang https://core.ac.uk/download/pdf/81983611.pdf – Tom Copeland Aug 25 '21 at 18:34
  • See pp. 74 and 75 of "A Koszul duality for props" by Vallette (https://arxiv.org/abs/math/0411542). – Tom Copeland Jun 04 '22 at 20:10

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