  Let $\pi_1$ be a standard representation of $\mathrm{GL}_{n+1}(F)$ and let
$\pi_2$ be the smooth dual of a standard representation of $\mathrm{GL}_n(F)$.
When $F$ is non-Archimedean, we prove that
$\mathrm{Ext}^i_{\mathrm{GL}_n(F)}(\pi_1, \pi_2)$ is $\cong \mathbb C$ when
$i=0$ and vanishes when $i \geq 1$. The main tool of the proof is a notion of
left and right Bernstein-Zelevinsky filtrations. An immediate consequence of
the result is to give a new proof on the multiplicity at most one theorem.
Along the way, we also study an application of an Euler-Poincar\'e pairing
formula of D. Prasad on the coefficients of Kazhdan-Lusztig polynomials.
  When $F$ is an Archimedean field, we use the left-right Bruhat-filtration to
prove a multiplicity result for the equal rank Fourier-Jacobi models of
standard principal series.
