On the distribution of the length of the second row of a young diagram under Plancherel measure

We investigate the probability distribution of the length of the second row of a Young diagram of size N equipped with Plancherel measure. We obtain a n expression for the generating function of the distribution in terms of a derivative of an associated Fredholm determinant, which can then be used to show that as N --> infinity the distribution converges to the Tracy-Widom distribution [TW1] for the second largest eigenvalue of a random GUE matrix . This paper is a sequel to [BDJ], where we showed that as N --> infinity t he distribution of the length of the first row of a Young diagram, or equiv alently, the length of the longest increasing subsequence of a random permu tation, converges to the Tracy-Widom distribution [TW1] for the largest eig envalue of a random GUE matrix.