A permutation of a set is an arrangement of its elements into a sequence and a
permutation statistic is a function from the union of all permutations of a
given set to the set of nonnegative integers. The study of permutation
statistics can be traced back to the work of M.P. McMahon in 1916. Since
then, the subject of permutation statistics has become quite active due to their
diverse connections to other mathematical areas such as symmetric functions,
basic hypergeometric series and random matrices.
For instance, we recently established a surprising connection between
statistics over members of the Fishburn family and transformation formulas
of basic hypergeometric series; we developed a two-stage saddle-point
approach to deal with generating functions with a sum-of-finite-product form.
As applications, a sequence of questions that are of great interest to the
combinatorics, topology and modular-form community have been addressed.
In this project, we will continue the research line on the interactions between
permutation statistics and generating functions. Our ultimate goals are to
establish new connections and to develop novel approaches via a
combination of fine combinatorial techniques and powerful analytic methods
to study likely behaviors of large random permutations, highlighting the
power of generating functions in bridging different areas such as
combinatorics, asymptotics and computer algebra.