Let $V=\Bbb{C}^3$ denote the standard representation of $\mathfrak{sl}_3(\Bbb{C})$. We can produce the irreducible representations of $\mathfrak{sl}_3(\Bbb{C})$ of a given highest weight by using the tensor powers $\mathrm{Sym}^a(V)\otimes \mathrm{Sym}^b(V^*).$
In Representation Theory, A First Course by Fulton and Harris, on p.184 there is a claim that in the representation $\mathrm{Sym}^a(V)\otimes \mathrm{Sym}^b(V^*)$ the weight multiplicities are given by $(i+1)(i+2)/2$ on the hexagon $H_i$ and given by the constant multiplicities $(b+1)(b+2)/2$ on the triangles $T_j$, as in the figure below: 
Here, $H_0$ denotes the outermost hexagon. There are thus two hexagons $H_0$ and $H_1$. The triangles then denote the $2$ inner layers, $T_0$ and $T_1$.
Being somewhat combinatorially impaired, I was hoping that someone might suggest to me a strategy for deriving these formulas.
I have tried working out some small examples, but the calculations become too immense before I can really see the pattern in a meaningful way. I was also thinking about using a character formula to prove this, but to no avail thus far.