  We consider the Random Euclidean Assignment Problem in dimension $d=1$, with
linear cost function. In this version of the problem, in general, there is a
large degeneracy of the ground state, i.e. there are many different optimal
matchings (say, $\sim \exp(S_N)$ at size $N$). We characterize all possible
optimal matchings of a given instance of the problem, and we give a simple
product formula for their number. Then, we study the probability distribution
of $S_N$ (the zero-temperature entropy of the model), in the uniform random
ensemble. We find that, for large $N$, $S_N \sim \frac{1}{2} N \log N + N s +
\mathcal{O}\left( \log N \right)$, where $s$ is a random variable whose
distribution $p(s)$ does not depend on $N$. We give expressions for the
asymptotics of the moments of $p(s)$, both from a formulation as a Brownian
process, and via singularity analysis of the generating functions associated to
$S_N$. The latter approach provides a combinatorial framework that allows to
compute an asymptotic expansion to arbitrary order in $1/N$ for the mean and
the variance of
