  The Mallows measure is measure on permutations which was introduced by
Mallows in connection with ranking problems in statistics. Under this measure,
the probability of a permutation $\pi$ is proportional to $q^{Inv(\pi)}$ where
$q$ is a positive parameter and $Inv(\pi)$ is the number of inversions in
$\pi$. We consider the length of the longest common subsequence (LCS) of two
independently permutations drawn according to $\mu_{n,q}$ and $\mu_{n,q'}$ for
some $q,q' >0$.
  We show that when $0<q,q'<1$, the limiting law of the LCS is Gaussian. In the
regime that $n(1-q) \to \infty$ and $n(1-q') \to \infty$ we show a weak law of
large numbers for the LCS. These results extend the results of \cite{Basu} and
\cite{Naya} showing weak laws and a limiting law for the distribution of the
longest increasing subsequence to showing corresponding results for the longest
common subsequence.
