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.