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Abstract: We introduce a new family of noncommutative analogues of the Hall–Littlewood symmetric functions. Our construction relies upon Tevlin''s bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall–Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explicit formula for: the steady state probability of each state in the partially asymmetric exclusion process (PASEP); the polynomial enumerating permutations with a fixed set of weak excedances according to crossings; the polynomial enumerating permutations with a fixed set of descent bottoms according to occurrences of the generalized pattern 2–31. [Copyright &y& Elsevier]

Abstract: We introduce a monoid structure on the set of binary search trees, by a process very similar to the construction of the plactic monoid, the Robinson–Schensted insertion being replaced by the binary search tree insertion. This leads to a new construction of the algebra of planar binary trees of Loday–Ronco, defining it in the same way as non-commutative symmetric functions and free symmetric functions. We briefly explain how the main known properties of the Loday–Ronco algebra can be described and proved with this combinatorial point of view, and then discuss it from a representation theoretical point of view, which in turns leads to new combinatorial properties of binary trees. [Copyright &y& Elsevier]

We present an overview of Λ-type operations on the algebra of quasi-symmetric functions. Nous présentons un survol de l'ensemble des propriétés de type Λ-anneau de l'algèbre des fonctions quasi-symétriques. [ABSTRACT FROM AUTHOR]