  A new equivalence relation on partitions is introduced to study the
isomorphism problem of invariant matrix algebras. For an algebraically closed
field $R$ of characteristic $p\ge 0$ and an element $\sigma$ in the symmetric
group $\Sigma_n$ of degree $n$, let $S_n(\sigma,R)$ be the $\sigma$-invariant
algebra of $n\times n$ matrices over $R$. Suppose $\sigma\in \Sigma_n$ and
$\tau\in \Sigma_m$ are of cycle types $\lambda=(\lambda_1,\cdots,\lambda_s)$
and $\mu=(\mu_1,\cdots,\mu_t)$, respectively, such that $p\nmid \lambda_i$ and
$p\nmid \mu_j$ for $1\le i\le s$ and $1\le j \le t$. We prove that $S_n(\sigma,
R)$ and $S_m(\tau, R)$ are isomorphic if and only if $m=n$ and both $\lambda$
and $\tau$ have the same partition polynomial. The two algebras are Morita
equivalent if and only if $d_{\lambda}\, \epsilon_{\lambda}(1) = d_{\mu}\,
\epsilon_{\mu}(1)$, where $\epsilon_{\lambda}(x)$ is the partition polynomial
of $\lambda$ and $d_{\lambda}$ stands for the greatest common divisor of
$\lambda_1,\cdots,\lambda_s$.
