Let be the group of permutations of the numbers (§26.2). With , is said to be an increasing subsequence of of length when . Let be the length of the longest increasing subsequence of . Then
where the distribution function is defined here by
and satisfies with and boundary conditions
where denotes the Airy function (§9.2).
The distribution function given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of Hermitian matrices; see Tracy and Widom (1994).