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The algebraic combinatorics of snakes. (English) Zbl 1246.05164
Summary: Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasi-symmetric functions. The results take the form of algebraic identities for type $$B$$ noncommutative symmetric functions, noncommutative supersymmetric functions and colored free quasi-symmetric functions.

##### MSC:
 05E05 Symmetric functions and generalizations 05E15 Combinatorial aspects of groups and algebras (MSC2010) 20F55 Reflection and Coxeter groups (group-theoretic aspects)
##### Keywords:
noncommutative symmetric functions; Euler numbers; snakes
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##### References:
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