**Title**: The Diagonals of Ferrers Diagrams

**Abstract**: In 1986, Garcia and Remmel defined the q-rook polynomials of Ferrers diagrams. They showed that they share many properties with the rook numbers introduced by Riordan and Kaplansky. In 1998, Haglund established connections between q-rook polynomials and matrices over finite fields.

In this talk, we reconstruct the theory of q-rook polynomials for Ferrers diagrams by focusing on the properties of their diagonals. We show that the diagonals define an equivalent relation on the set of Ferrers diagram. As a consequence, we provide results on constructions of linear spaces of matrices satisfying the Etzion-Siberstein Conjecture and we establish connection with the problem of counting matrices of given rank supported on a Ferrers diagram.

The new results in this talk are joint work with A. Gruica and A. Ravagnani.