  Introduced by Mallows in statistical ranking theory, Mallows permutation
model is a class of non-uniform probability measures on the symmetric group
$S_n$ that depend on a distance metric $d(\sigma,\tau)$ on $S_n$ and a scale
parameter $\beta$. Taking the distance metric to be the $L^1$ and $L^2$
distances--which are respectively known as Spearman's footrule and Spearman's
rank correlation in the statistics literature--leads to Mallows permutation
models with $L^1$ and $L^2$ distances.
  In this paper, we study the length of the longest increasing subsequence of
random permutations drawn from Mallows permutation models with $L^1$ and $L^2$
distances. For both models and various regimes of the scale parameter $\beta$,
we determine the typical order of magnitude of the length of the longest
increasing subsequence and establish a law of large numbers for this length.
For Mallows permutation model with the $L^1$ distance, when $\beta\sim \theta
n^{-1}$ for some fixed $\theta>0$, the typical length of the longest increasing
subsequence is of order $\sqrt{n}$; when $n^{-1}\ll \beta \ll 1$, this typical
length is of order $n\sqrt{\beta}$. For Mallows permutation model with the
$L^2$ distance, when $\beta\sim \theta n^{-2}$ for some fixed $\theta >0$, the
typical length of the longest increasing subsequence is of order $\sqrt{n}$;
when $n^{-2}\ll \beta \ll 1$, this typical length is of order
$n\beta^{\frac{1}{4}}$.
