Multiplicity formulas for fundamental strings of representations of classical Lie algebras

We call the \emph{$p$-fundamental string} of a complex simple Lie algebra to the sequence of irreducible representations having highest weights of the form $k{\omega }_1+{\omega }_p$ for $k\ge 0$, where ${\omega }_j$ denotes the $j$-th fundamental weight of the associated root system. For a classical complex Lie algebra, we establish a closed explicit formula for the weight multiplicities of any representation in any $p$-fundamental string.