  The study of longest increasing subsequences (LIS) in permutations led to
that of Young diagrams via Robinson-Schensted's (RS) correspondence. In a
celebrated paper, Vershik and Kerov obtained a limit theorem for such diagrams
and found that the LIS of a uniform permutation of size n behaves as
$2\sqrt{n}$. Independently and much later, Hoppen et al. introduced the theory
of permutons as a scaling limit of permutations. In this paper, we extend in
some sense the RS correspondence of permutations to the space of permutons.
When the "RS-tableaux" of a permuton are non-trivial, we show that the
RS-tableaux of random permutations sampled from this permuton exhibit a linear
behavior, in the sense that their first rows and columns have lengths of linear
order. In particular, the LIS of such permutations behaves as a multiple of n.
We also prove some large deviation results for these convergences. Finally, by
studying asymptotic properties of Fomin's algorithm for permutations, we show
that the RS-tableaux of a permuton satisfy a partial differential equation.
