partition. Furthermore, if $(st-s-t)-(k's-l's)$ is a first-column hook length for some $(s,t)$-core partition $\lambda$, then so is $(st-s-t)-(ks-lt)$ for all $k>k'$ and $l>l'$.

In particular, Lemma \[andy\] implies a correspondence between $(s,t)$-cores a family of monotonic paths in a certain $s\times t$ lattice. \[A lattice path is [*monotonic*]{} if it only moves one position up or one position to the right at every step.\] The lattice and paths are constructed in the following way. The left-and-topmost box, with upper-left corner labeled $(0,s)$, contains the value $st-s-t$. One horizontal position right from $(0,s)$ is the box whose upper left corner is labeled $(1,s)$, and the value in that box is $st-2s-t$ (a reduction by $s$). One vertical position down from $(0,s)$ is the box with label $(0,s-1)$; the value inside is $st-s-2t$. Complete the lattice coordinates with their values in this fashion. Now consider a lattice path $\pi$ from the bottom left-most corner position labeled $(0,0)$ to the top right-most corner position labeled $(t,s)$ that stays above and does not cross the border between the positive and negative values (the $\pm$[*[border]{}*]{}). Then, by construction, the values trapped between $\pi$ and the $\pm$border will all be positive, and are exactly the first-column 