Seaweed algebras and the unimodal spectrum property

If $\mathfrak{g}$ is a Frobenius Lie algebra, then the spectrum of $\mathfrak{g}$ is an algebraic invariant equal to the multiset of eigenvalues corresponding to a particular operator acting on $\mathfrak{g}$. In the case of Frobenius seaweed subalgebras of $A_{n-1}=\mathfrak{sl}(n)$, or type-A seaweeds for short, it has been shown that the spectrum can be computed combinatorially using an attendant graph. With the aid of such graphs, it was further shown that the spectrum of a type-A seaweed consists of an unbroken sequence of integers centered at $\frac{1}{2}$. It has been conjectured that if the eigenvalues are arranged in increasing order, then the sequence of multiplicities forms a unimodal sequence about $\frac{1}{2}$. Here, we establish this conjecture for certain families of Frobenius type-A seaweeds by finding explicit formulas for their spectra; in fact, for some families we are able to show that the corresponding sequences of multiplicities form log-concave sequences. All arguments are combinatorial.