If I understood your question correctly, the numbers you're looking for are called Ballot numbers. The number of paths from $(0,0)$ to $(m,n)$ (where $m>n$) which stay below the diagonal is $\frac{m-n}{m+n}\binom{m+n}{m}$.
Moreover, if $m>r \cdot n$, then the number of lattice paths from $(0,0)$ to $(m,n)$ which stay below the line $x=r\cdot y$ is $\frac{m-rn}{m+n}\binom{m+n}{m}$. (I haven't worked this out, but Ira Gessel says so in Introduction to Lattice Path Enumeration)
