A uniformly chosen permutation of \(\{1,2,3,\dots,n\}\) has a longest increasing (or decreasing) subsequence of length about \(2n^{1/2}\), and the fluctuations around this are of order \(n^{1/6}\). In this talk, we will consider permutations whose longest increasing subsequence is much shorter than that. In the case where the longest increasing subsequence is of constant order, we show that the appropriately scaled limit of the permutation is given by the eigenvalue process for an ensemble of random matrices. If the longest increasing subsequence has length \(n^a\) for \(a<1/2\), then we think that the limiting object will be the Brownian watermelon.