# §32.14 Combinatorics

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

32.14.3,
32.14.4,

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).

See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.