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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)

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)

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⋅n, then the number of lattice paths from (0,0) to (m,n) which stay below the line x=r⋅y is \fra{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)

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)

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}

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⋅n, then the number of lattice paths from (0,0) to (m,n) which stay below the line x=r⋅y is \fra{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)