Geodesic
Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 66 (2011) no. 2, pp. 271-367

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This survey is devoted to the classical problem of flag numbers of convex polytopes, and contains an exposition of results obtained on the basis of connections between the theory of convex polytopes and a number of modern directions of research. Bibliography: 62 titles.
Keywords: flag numbers, flag polynomials, Leibniz–Hopf algebra, Lyndon words, $\boldsymbol{cd}$-index.
Mots-clés : Dehn–Sommerville relations, universal $G$-polynomial 
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V. M. Buchstaber; N. Yu. Erokhovets. Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 66 (2011) no. 2, pp. 271-367. http://geodesic.mathdoc.fr/item/RM_2011_66_2_a2/