Counting north east lattice paths in a rhomboid

Question

I am counting the North East (i.e. (0,1) and(1,0)) lattice paths inside some polygons. I managed to count the paths inside a triangle (Catalan's numbers) and inside a trapezoid (Bertrand's ballot). I am counting now inside a rhomboid, from end to end. More precisely, the number of north east paths from the origin to (m,n-a) that do not go above the line x=y nor below the line y=x-a.

Answer 1

I found what I needed. See enumerating lattice paths articles