Permutations with short monotone subsequences

We consider permutations of 1,2,…,n2 whose longest monotone subsequence is of length n and are therefore extremal for the Erdős–Szekeres theorem. Such permutations correspond via the Robinson–Schensted correspondence to pairs of square n×n Young tableaux. We show that all the bumping sequences are constant and therefore these permutations have a simple description in terms of the pair of square tableaux. We deduce a limit shape result for the plot of values of the typical such permutation, and other properties of these extremal permutations.