= Dominant weight multiplicity formula for sl3
{title2=$m_{a,b}(p,q)=\max\{0,1+\min(a,b,r,s)\}$}
For the irreducible <highest-weight representation> of the <special linear Lie algebra> $\mathfrak{sl}_3$ with <Dynkin labels> $(a,b)$, let $(p,q)$ be a <dominant integral weight>. Put $r=(2a+b-2p-q)/3$ and $s=(a+2b-p-2q)/3$. If these are not <integers>, the <weight multiplicity> is zero; otherwise it is $\max\{0,1+\min(a,b,r,s)\}$.
To derive the formula, use the <sl3 interlacing character formula> with top row $(a+b,b,0)$. The target diagonal exponents are $(s+p+q,s+q,s)$, and an interlacing middle row $(P,Q)$ has $P+Q=2s+p+2q$, with bottom entry $s+p+q$. Thus
$$
\max\{b,2s+p+2q-b,s+p+q\}\leq P\leq\min\{a+b,2s+p+2q\}.
$$
Every integral $P$ in this interval gives exactly one <Gelfand–Tsetlin basis> vector. The upper bound minus the lower bound is the minimum of six differences: $a,s,r,r+p+q,b,s+q$. Since $p,q\geq0$, this minimum reduces to $\min(a,b,r,s)$. Counting the interval proves the formula, including absent <weights of a representation>. This converts a two-dimensional <weight diagram> calculation into four elementary boundary distances.
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