KöMaL 2023 A851

Let $k$, $l$ and $m$ be positive integers. Let $ABCDEF$ be a hexagon that has a center of symmetry whose angles are all $120^{\circ}$ and let its sidelengths be $AB=k$, $BC=l$ and $CD=m$. Let $f(k,l,m)$ denote the number of ways we can partition hexagon $ABCDEF$ into rhombi with unit sides and an angle of $120^{\circ}$. Prove that by fixing $l$ and $m$, there exists polynomial $g_{l,m}$ such that $f(k,l,m)=g_{l,m}(k)$ for every positive integer $k$, and find the degree of $g_{l,m}$ in terms of $l$ and $m$.