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Binary shuffle bases for quasi-symmetric functions. (English) Zbl 1335.05183
Summary: We construct bases of quasi-symmetric functions whose product rule is given by the shuffle of binary words, as for multiple zeta values in their integral representations, and then extend the construction to the algebra of free quasi-symmetric functions colored by positive integers. As a consequence, we show that the fractions introduced in L. Guo and B. Xie [Ramanujan J. 25, No. 3, 307–317 (2011; Zbl 1222.11108)] provide a realization of this algebra by rational moulds extending that of free quasi-symmetric functions given in F. Chapoton et al. [Int. Math. Res. Not. 2008, Article ID rnn018, 22 p. (2008; Zbl 1146.18301)].

MSC:
05E05 Symmetric functions and generalizations
16T30 Connections of Hopf algebras with combinatorics
11M32 Multiple Dirichlet series and zeta functions and multizeta values
11M41 Other Dirichlet series and zeta functions
18D50 Operads (MSC2010)
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