Motivation:
I'm interested in understanding the role that noncrossing partitions play in Hopf algebras/monoids (HAs) as the components of the power series of the compositional inverse of formal power series (or analytic functions at the origin, cf. OEIS A134264). The roles in HAs of the associahedra and permutahedra as delineating the combinatorics of compositional and multiplicative inversion are in encoding the antipodes in some HAs. I'm parttcularly interested in unwrapping this statement in Ref. 6 (Varilly et al.):
Use of partitions with special properties may lead to other incidence algebras: for instance, if we restrict to noncrossing partitions, we obtain a cocommutative Hopf algebra, with the commutative group operation on characters essentially corresponding to Lagrange reversion of the Cauchy product of reverted series [cf. Stanley, Enumerative Comb., Vol. 2].
Request:
To this end, I would like to understand the lingo and constructs of HAs at an elementary, intuitive level and am seeking free pdf papers or books on the Web that discuss HAs at increasing levels of generality with concrete examples.
Some that I'm already aware of are (in roughly increasing generality)
1) "Trees, renormalization, and diffetential equations" by c. Brouder
2) "A very basic introduction to Hopf algebras" by Selig
3) "Coalgebras and Bialgebras in Combinatorics" by S. A. Joni G.‐C. Rota
4) "Combinatorial Route to Algebra: The Art of Composition & Decomposition" by Blasiak
5) "Hopf Algebras in General and in Combinatorial Physics: a practical introduction" by Duchamp, Blasiak, Horzela, Penson, and Solomon
6) "Faa di Bruno Hopf algebras" by Figueroa, Gracia-Bondia and Varilly
7) "Combinatorial Hopf algebras in quantum field theory I" by Figueroa and Gracia-Bondia