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Let $F$ be a non-archimedean local field. Let $\Pi$ be a principal seriesrepresentation of $\rm{GL}_n(F)$ induced from a cuspidal representation of aLevi subgroup. When $\pi$ is an essentially square integrable representation of$\rm{GL}_{n-1}(F)$ we prove that $\rm{Hom}_{\rm{GL}_{n-1}}(\Pi,\pi) =\mathbb{C}$ and $\rm{Ext}^i_{\rm{GL}_{n-1}}(\Pi,\pi) = 0$ for all integers$i\geq 1$, with exactly one exception (up to twists), namely, when $\Pi=\nu^{-(\frac{n-1}{2})} \times \nu^{-(\frac{n-3}{2})} \times \ldots \times\nu^{(\frac{n-1}{2})}$ and $\pi$ is the Steinberg. When $\Pi=\nu^{-(\frac{n-1}{2})} \times \nu^{-(\frac{n-3}{2})} \times \ldots \times\nu^{(\frac{n-1}{2})}$ and $\pi$ is the Steinberg of $\rm{GL}_{n-1}(F)$, then$\dim \rm{Hom}_{\rm{GL}_{n-1}(F)}(\Pi,\pi)=n$. We also exhibit specificprincipal series for which each of the intermediate multiplicities $2, 3,\cdots, (n-1)$ are attained.  Along the way, we also give a complete list of those irreducible non-genericrepresentations of $\rm{GL}_{n}(F)$ that have the Steinberg of$\rm{GL}_{n-1}(F)$ as a quotient upon restriction to $\rm{GL}_{n-1}(F)$. Wealso show that there do not exist non-generic irreducible representations of$\rm{GL}_{n}(F)$ that have the generalized Steinberg as a quotient uponrestriction to $\rm{GL}_{n-1}(F)$.

On Higher Multiplicity upon Restriction from $\rm{GL}(n)$ to $\rm{GL}(n-1)$