Research Project:Counting matrices over finite fields of given rank satisfying some restrictions on their entries is a classical theme which has been investigated from several points of view: geometric, combinatorial, probabilistic, and others. Enumeration of Coxeter groups according to so-called permutation statistics is an interesting topic with many applications to other areas of mathematics. In particular, recent results demonstrate that permutation statistics can be employed in the enumeration of matrices with suitable constraints on their entries and their ranks. The aim of this project is to investigate the remarkable (but presently mysterious) connection between such enumerative problems for matrices on the one hand and permutation statistics on hyperoctahedral groups on the other hand. During my stay at ESI, I will work towards establishing a conceptual relationship between the two objects under examination and exploit it to obtain structural and enumerative results in both areas. I will also use my results to study, by means of appropriate generating functions, the asymptotic behaviour of certain families of matrices.