  The Mallows measure on the symmetric group $S_n$ is the probability measure
such that each permutation has probability proportional to $q$ raised to the
power of the number of inversions, where $q$ is a positive parameter and the
number of inversions of $\pi$ is equal to the number of pairs $i<j$ such that
$\pi_i > \pi_j$. We prove a weak law of large numbers for the length of the
longest increasing subsequence for Mallows distributed random permutations, in
the limit that $n$ tends to infinity and $q$ tends to 1 in such a way that
$n(1-q)$ has a limit in $\R$.
