Permutations with Verma Multiplicities $[M(p):L(q)] \geq 2$

We consider permutations $q$ in the symmetric group $S_n$ whose Verma multiplicities in the principal block of $\mathcal{O}(\mathfrak{sl}_n)$ satisfy $[M(p):L(q)] \geq 2$. We present a construction along with a diagrammatic visualization, showing how permutations in $S_n$ with this property generate a family of permutations in $S_{n+1}$ that share the same multiplicity property. While the method does not recover all such permutations in $S_{n+1}$, it systematically generates many new examples. In addition, we present all permutations in $S_5$ with $[M(\mathrm{id}):L(q)] \geq 2$ in a Bruhat diagram and describe all permutations in $S_6$ and $S_7$ with non-simple Verma multiplicities.