  Let $\mathfrak{g}$ be a classical complex simple Lie algebra. Let
$L(\lambda)$ be a highest weight module of $\mathfrak{g}$ with highest weight
$\lambda-\rho$, where $\rho$ is half the sum of positive roots. The associated
variety of the annihilator ideal of $L(\lambda)$ is called the annihilator
variety of $L(\lambda)$.It is known that the annihilator variety of any highest
weight module $L(\lambda)$ is the Zariski closure of a nilpotent orbit in
$\mathfrak{g}^*$. But in general, this nilpotent orbit is not easy to describe
for a given highest weight module $L(\lambda)$. In this paper, we will give
some simple formulas to characterize this unique nilpotent orbit appearing in
the annihilator variety of a highest weight module for classical Lie algebras.
Our formulas are given by introducing two algorithms, i.e., bipartition
algorithm and partition algorithm. To get a special or metaplectic special
partition from a domino type partition, we define the H-algorithm based on the
Robinson-Schensted insertion algorithm. By using this H-algorithm, we can
easily determine this nilpotent orbit from the information of $\lambda$.
