  Let $F$ be a non-archimedean local field. Let $\Pi$ be a principal series
representation of $\rm{GL}_n(F)$ induced from a cuspidal representation of a
Levi 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 specific
principal 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-generic
representations of $\rm{GL}_{n}(F)$ that have the Steinberg of
$\rm{GL}_{n-1}(F)$ as a quotient upon restriction to $\rm{GL}_{n-1}(F)$. We
also show that there do not exist non-generic irreducible representations of
$\rm{GL}_{n}(F)$ that have the generalized Steinberg as a quotient upon
restriction to $\rm{GL}_{n-1}(F)$.
