A generalization of Eulerian numbers via rook placements

We consider a generalization of Eulerian numbers which count the number of placements of cn “rooks” on an n× n board where there are exactly c rooks in each row and each column, and exactly k rooks below the main diagonal. The standard Eulerian numbers correspond to the case c = 1. We show that for any c the resulting numbers are symmetric and give generating functions of these numbers for small values of k.