  We study the entropy $S$ of longest increasing subsequences (LIS), i.e., the
logarithm of the number of distinct LIS. We consider two ensembles of
sequences, namely random permutations of integers and sequences drawn i.i.d.\
from a limited number of distinct integers. Using sophisticated algorithms, we
are able to exactly count the number of LIS for each given sequence.
Furthermore, we are not only measuring averages and variances for the
considered ensembles of sequences, but we sample very large parts of the
probability distribution $p(S)$ with very high precision. Especially, we are
able to observe the tails of extremely rare events which occur with
probabilities smaller than $10^{-600}$. We show that the distribution of the
entropy of the LIS is approximately Gaussian with deviations in the far tails,
which might vanish in the limit of long sequences. Further we propose a
large-deviation rate function which fits best to our observed data.
