The compositional inversions noted in the Oct. 9-th entry "Flipping
Functions with Permutohedra" have counterparts with respect to the
associahedra, or Stasheff polytopes. The h-polynomials of the
associahedra (see the Narayana number triangle OEIS-A001263) can be
refined as OEIS-A134264 for the coefficients of an indirect Lagrange
method for determining the compositional inverse of a function,
whereas the corresponding f-polynomials (see OEIS-A033282) can be
refined as the unsigned polynomials in OEIS-A133437 (or A111785) for
the coefficients of a Lagrange inversion for an ordinary generating
function. (OEIS denotes the Online Encyclopedia of Integer Sequences.)

Flipping Functions with Permutohedra A short note on forming the multiplicative and compositional inverses of functions using the refined Eulerian h-polynomials and refined face polynomials of permutohedra.

The notes Combinatorics of OEIS-A074060 sketch how combinatorial interpretations of LaGrange inversion and the 2-restricted Stirling numbers of the first kind provide a combinatorial construction for the array OEIS-A074060 (Graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus 0 satisfying the associativity equations of physics).

 Short Note on Lagrange Inversion is an addendum to OEIS-A134685.

The set of notes Mathemagical Forests is an expansion of the May notes and discusses some connections between rooted trees, derivative operators, Lagrange inversion, the Legendre transformation, the Faa di Bruno formula, Sheffer sequences and umbral calculus, and the infinite Witt Lie algebra.

In the last several months, I've been exploring the connections between umbral calculus and integer sequences and have published several items, searchable under the name Tom Copeland, in the Online Encyclopedia of Integer Sequences (OEIS). During the process, I became aware of the ubiquitousness of trees in combinatorics and found it useful to go back to the origins to get a good feel for them. The resulting set of notes "A Walk in the Woods with Cayley and Comtet " might be a helpful introduction to others on some elementary relations between the infinitesimal calculus and rooted trees ala Cayley.