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.