  If $\mathfrak{g}$ is a Frobenius Lie algebra, then for certain $F\in
\mathfrak{g}^*$ the natural map $\mathfrak{g}\longrightarrow \mathfrak{g}^* $
given by $x \longmapsto F[x,-]$ is an isomorphism. The inverse image of $F$
under this isomorphism is called a principal element. We show that if
$\mathfrak{g}$ is a Frobenius seaweed subalgebra of $A_{n-1}=\mathfrak{sl}(n)$
then the spectrum of the adjoint of a principal element consists of an unbroken
set of integers whose multiplicities have a symmetric distribution. Our proof
methods are constructive and combinatorial in nature.
