  We study the distribution of the length of longest increasing subsequences in
random permutations of $n$ integers as $n$ grows large and establish an
asymptotic expansion in powers of $n^{-1/3}$. Whilst the limit law was already
shown by Baik, Deift and Johansson to be the GUE Tracy-Widom distribution $F$,
we find explicit analytic expressions of the first few finite-size correction
terms as linear combinations of higher order derivatives of $F$ with rational
polynomial coefficients. Our proof replaces Johansson's de-Poissonization,
which is based on monotonicity as a Tauberian condition, by analytic
de-Poissonization of Jacquet and Szpankowski, which is based on growth
conditions in the complex plane; it is subject to a tameness hypothesis
concerning complex zeros of the analytically continued Poissonized length
distribution. In a preparatory step an expansion of the hard-to-soft edge
transition law of LUE is studied, which is lifted into an expansion of the
Poissonized length distribution for large intensities. Finally, expansions of
Stirling-type approximations and of the expected value and variance of the
length distribution are given.
