On increasing subsequences of minimal Erdös-Szekeres permutations

Let π be a minimal Erdös-Szekeres permutation of 1, 2, ..., n2, and let ln,k be the length of the longest increasing subsequence in the segment (π(1), ..., π(k)). Under uniform measure we establish an exponentially decaying bound of the upper tail probability for ln,k, and as a consequence we obtain a complete convergence, which is an improvement of Romik’s recent result. We also give a precise lower exponential tail for ln,k.