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Introduced by Mallows in statistical ranking theory, Mallows permutationmodel 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 scaleparameter $\beta$. Taking the distance metric to be the $L^1$ and $L^2$distances--which are respectively known as Spearman's footrule and Spearman'srank correlation in the statistics literature--leads to Mallows permutationmodels with $L^1$ and $L^2$ distances.  In this paper, we study the length of the longest increasing subsequence ofrandom 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 longestincreasing subsequence and establish a law of large numbers for this length.For Mallows permutation model with the $L^1$ distance, when $\beta\sim \thetan^{-1}$ for some fixed $\theta>0$, the typical length of the longest increasingsubsequence is of order $\sqrt{n}$; when $n^{-1}\ll \beta \ll 1$, this typicallength is of order $n\sqrt{\beta}$. For Mallows permutation model with the$L^2$ distance, when $\beta\sim \theta n^{-2}$ for some fixed $\theta >0$, thetypical length of the longest increasing subsequence is of order $\sqrt{n}$;when $n^{-2}\ll \beta \ll 1$, this typical length is of order$n\beta^{\frac{1}{4}}$.

The length of the longest increasing subsequence of Mallows permutation models with $L^1$ and $L^2$ distances