Question: Given positive, coprime integers $m<n$, consider the Young diagram $Y$ formed by the lattice points in the Cartesian plane lying below the line from $(0,0)$ to $(m,n)$ and within the rectangle $(0,m)\times (0,n)$. The case $m=5$, $n=7$ is shown below. For positive integer $x$, how many integer partitions of $x$ are there whose Young diagram is $\leq Y$ in the Young lattice?
Equivalently, how many lattice paths are there from $(0,0)$ to $(m,n)$, moving only up and right and lying below the diagonal, whose integral over $[0,m]$ is precisely $x$?
This problem originated from this MSE question. The problem considered in that question was the following: Given positive integers $m<n$ (not necessarily coprime), how many binary sequences $(s_j)_{j=0}^\infty$ with exactly $x$ 1's are there which have the property that $s_j=0 \implies s_{j+m}=0 \land s_{j+n}=0$. That problem can be reduced to the one I give above.
The problem posed above seems at least as hard as enumerating integer partitions, since for $x<m$ the number of such Young diagrams/lattice paths is precisely the number of integer partitions of $x$. Probably the best I could hope for is a convenient expression for the generating function.
