A permutation group G (acting on a set Ω, usually infinite) is said to be oligomorphic if G has only finitely many orbits on Ω n (the set of n-tuples of elements of Ω). Such groups have traditionally been linked with model theory and combinatorial enumeration; more recently their group-theoretic properties have been studied, and links with graded algebras, Ramsey theory, topological dynamics, and other areas have emerged. This paper is a short summary of the subject, concentrating on the enumerative and algebraic aspects but with an account of grouptheoretic properties. The first section gives an introduction to permutation groups and to some of the more specific topics we require, and the second describes the links to model theory and enumeration. We give a spread of examples, describe results on the growth rate of the counting functions, discuss a graded algebra associated with an oligomorphic group, and finally discuss group-theoretic properties such as simplicity, the small index property, and “almost-freeness”.

Abstract. We relate a universal formula for the deformation quantization of Poisson structures (⋆-products) on R d proposed by Maxim Kontsevich to the Campbell-Baker-Hausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for such formulae. For the dual of a Lie algebra, the ⋆-product given by the universal enveloping algebra via symmetrization is shown to be of this type. In fact this ⋆-product is essentially given by the Campbell-Baker-Hausdorff (CBH) formula. We call this the CBH-quantization. Next we limn Kontsevich’s construction using a graphical representation for differential calculus. We outline a structure theory for the weighted graphs which encode bi-differential operators in his formula and compute certain weights. We then establish that the Kontsevich and CBH quantizations are identical for the duals of nilpotent Lie algebras. Consequently part of Kontsevich’s ⋆-product is determined by the CBH formula. Working the other way, we have a graphical encoding for the

Abstract. Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication ∗ on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product x. We extend this commutative algebra structure to a Hopf algebra (A, ∗, ∆); in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, the quasi-shuffle product is in fact no more general than the shuffle product; we give an isomorphism exp of the shuffle Hopf algebra (A,x,∆) onto (A, ∗, ∆). Both the set L of Lyndon words on A and their images {exp(w) | w ∈ L} freely generate the algebra (A, ∗). We also consider the graded dual of (A, ∗,∆). We define a deformation ∗q of ∗ that coincides with ∗ when q = 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasi-symmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums.

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Noncommutative analogues of classical operations on symmetric functions are investigated, and applied to the description of idempotents and nilpotents in descent algebras. Its is shown that any sequence of Lie idempotents (one in each descent algebra) gives rise to a complete set of indecomposable orthogonal idempotents of each descent algebra, and various deformations of the classical sequences of Lie idempotents are obtained. In particular, we obtain several q-analogues of the Eulerian idempotents and of the GarsiaReutenauer idempotents.

We present representation theoretical interpretations of quasi-symmetric functions and noncommutative symmetric functions in terms of quantum linear groups and Hecke algebras at q = 0. We obtain in this way a noncommutative realization of quasi-symmetric functions analogous to the plactic symmetric functions of Lascoux and Schützenberger. The generic case leads to a notion of quantum Schur functions.

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A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type B, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra.

this paper we will develop a connection between parking functions and another topic, viz., noncrossing partitions

This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple q-analogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in theoretical physics (the quon algebra).