  We study numerically the distributions of the length $L$ of the longest
increasing subsequence (LIS) for the two cases of random permutations and of
one-dimensional random walks. Using sophisticated large-deviation algorithms,
we are able to obtain very large parts of the distribution, especially also
covering probabilities smaller than $P(L) = 10^{-1000}$. This enables us to
verify for the length of the LIS of random permutations the analytically known
asymptotics of the rate function and even the whole Tracy-Widom distribution,
to which we observe a rather fast convergence in the larger than typical part.
For the length $L$ of LIS of random walks, where no analytical results are
known to us, we test a proposed scaling law and observe convergence of the
tails into a collapse for increasing system size. Further, we obtain estimates
for the leading order behavior of the rate functions of both tails.
