My research involves the study of interesting interactions between algebraic structures (spaces of harmonic polynomials, representations of reflection groups, etc.) and combinatorial objects (trees, integer partitions, permutations, Catalan structures, parking functions, etc.). These interactions give rise to several identities, often expressed in terms of generating functions or symmetric functions (Schur functions, Macdonald polynomials, etc.).
Most recently, I have been collaborating with Adriano Garsia, Mark Haiman and others on algebraic and combinatorial aspects of the study of (m,n)-Parking functions. This is a continuation (with a lot of new twists) of our longstanding study of combinatorial aspects of Diagonal Harmonic Polynomials of the Symmetric Group.
In collaboration with other members of Lacim (mainly Pierre Leroux and Gilbert Labelle), I contributed in the 80-90 to the original development of the Theory of Species (introduced by André Joyal). For a rapid introduction to the theory, see Combinatorial Species. My collaboration with Simon Plouffe has contributed to the emergence of tools such as GFUN (in Maple).