  Two-dimensional Yang-Mills theory is a useful model of an exactly solvable
gauge theory with a string theory dual at large $N$. We calculate entanglement
entropy in the $1/N$ expansion by mapping the theory to a system of $N$
fermions interacting via a repulsive entropic force. The entropy is a sum of
two terms: the "Boltzmann entropy", $\log \dim (R)$ per point of the entangling
surface, which counts the number of distinct microstates, and the "Shannon
entropy", $- \sum p_R \log p_R$, which captures fluctuations of the macroscopic
state. We find that the entropy scales as $N^2$ in the large $N$ limit, and
that at this order only the Boltzmann entropy contributes. We further show that
the Shannon entropy scales linearly with $N$, and confirm this behaviour with
numerical simulations. While the term of order $N$ is surprising from the point
of view of the string dual - in which only even powers of $N$ appear in the
partition function - we trace it to a breakdown of large $N$ counting caused by
the replica trick. This mechanism could lead to corrections to holographic
entanglement entropy larger than expected from semiclassical field theory.
