  Considered are the large $N$, or large intensity, forms of the distribution
of the length of the longest increasing subsequences for various models.
Earlier work has established that after centring and scaling, the limit laws
for these distributions relate to certain distribution functions at the hard
edge known from random matrix theory. By analysing the hard to soft edge
transition, we supplement and extend results of Baik and Jenkins for the
Hammersley model and symmetrisations, which give that the leading correction is
proportional to $z^{-2/3}$, where $z^2$ is the intensity of the Poisson rate,
and provides a functional form as derivates of the limit law. Our methods give
the functional form both in terms of Fredholm operator theoretic quantities,
and in terms of Painlev\'e transcendents. For random permutations and their
symmetrisations, numerical analysis of exact enumerations and simulations gives
compelling evidence that the leading corrections are proportional to
$N^{-1/3}$, and moreover provides an approximation to their graphical forms.
