  This paper is to establish a natural connection between regular
representations for a vertex operator algebra $V$ and $A_{n}(V)$-$A_{m}(V)$
bimodules of Dong and Jiang. Let $W$ be a weak $V$-module and let $(m,n)$ be a
pair of nonnegative integers. We study two quotient spaces
$A_{n,m}^{\dagger}(W)$ and $A^{\diamond}_{n,m}(W)$ of $W$.
  It is proved that the dual space $A^{\dagger}_{n,m}(W)^{*}$ viewed as a
subspace of $W^*$ coincides with the level-$(m,n)$ vacuum subspace of the
regular representation module $\mathfrak{D}_{(-1)}(W)$. By making use of this
connection, we obtain an $A_{n}(V)$-$A_m(V)$ bimodule structure on both
$A_{n,m}^{\dagger}(W)$ and $A^{\diamond}_{n,m}(W)$. Furthermore, we obtain an
$\N$-graded weak $V$-module structure together with a commuting right
$A_m(V)$-module structure on $A^{\diamond}_{\Box,m}(W):=\oplus_{n\in
\N}A^{\diamond}_{n,m}(W)$. Consequently, we recover the corresponding results
and roughly confirm a conjecture of Dong and Jiang.
