Some Hopf algebras encode relevant combinatorial information and are quite common and useful in algebraic combinatorics? in general. More recently Hopf algebras appeared also in the study of renormalization of QFT, controlling its combinatorics. Combinatorial Hopf algebra may denote either informal notion of Hopf algebra coming from combinatorics (and, in particular, study of various rings of symmetric functions), but also, now standard notion as a pair of a Hopf algebra over equipped with a character , the zeta function of a Hopf algebra.
Federico Ardila, Hopf algebras and combinatorics, 30 lectures, youtube list
There is a classical survey
Gian-Carlo Rota, Hopf algebra methods in combinatorics, Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), pp. 363–365, Colloq. Internat. CNRS 260, CNRS, Paris 1978.
and for the basic intuition for Hopf algebras in ocmbinatorics see the chapter 5 of S. Majid’s Foundation of quantum group theory. Nowdays the major book on combinatorial Hopf algebras is
G-C. Rota, J. A. Stein, Plethystic Hopf algebras, Proc. Nat. Acad. Sci. U.S.A. 91 (1994), no. 26, 13057–13061, MR96e:16054, Plethystic algebras and vector symmetric functions, Proc. Nat. Acad. Sci. U.S.A. 91 (1994), no. 26, 13062–13066, MR96e:16055
Kurusch Ebrahimi-Fard, Combinatorial Hopf algebras+, lectures at CIMAT summer school, slides pdf
F. Hivert, J.-C. Novelli, J.-Y. Thibon, Trees, functional equations and combinatorial Hopf algebras, Europ. J. Comb. 29 (1) (2008), 1682–1695.
Mercedes H. Rosas, Gian-Carlo Rota, Joel Stein, A combinatorial overview of the Hopf algebra of MacMahon symmetric functions, Ann. Comb. 6 (2002), no. 2, 195–207.
Bertfried Fauser, On the Hopf algebraic origin of Wick normal ordering, J. Phys. A 34 (2001), no. 1, 105–115
Damien Calaque, Kurusch Ebrahimi-Fard, Dominique Manchon, Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series, Adv. in Appl. Math. 47 (2011), no. 2, 282–308
K. Ebrahimi-Fard, Li Guo, Mixable shuffles, quasi-shuffles and Hopf algebras, J. Algebraic Combin. 24 (2006), no. 1, 83–101, MR2007d:05152,doi
K. Ebrahimi-Fard, D. Kreimer, The Hopf algebra approach to Feynman diagram calculations, J. Phys. A 38 (2005), no. 50, R385–R407, MR2006k:81266, doi