Fulton and Harris describe the weight lattice of the $\mathfrak{sl}_3(\mathbb{C})$ representation (for $a \geq b$) $\text{Sym}^a(V) \otimes \text{Sym}^b(V^*)$ as a sequence of $b$ shrinking concentric hexagons $H_i$ with vertices at $(a-i)L_3 -(b-i)L_3$ followed by a sequence of triangles $T_j$ with vertices at the points $(a-b-3j)L_1$. They go on to claim
An examination of the representation $\text{Sym}^a(V) \otimes \text{Sym}^b(V^*)$ shows that it has multiplicity $(i+1)(i+2)/2$ on the hexagon $H_i$, and then a constant multiplicity of$(b+1)(b+2)/2$ on all the triangles.
Can someone tell me why an examination of the representation gives such multiplicities? I can't seem to figure it out. I spent some time trying to derive this as the number of sums of eigenvalues in each of the factors in the tensor products that vies particular sums, with no real luck. What the trick to this?