  Given two random finite sequences from $[k]^n$ such that a prefix of the
first sequence is a suffix of the second, we examine the length of their
longest common subsequence. If $\ell$ is the length of the overlap, we prove
that the expected length of an LCS is approximately $\max(\ell,
\mathbb{E}[L_n])$, where $L_n$ is the length of an LCS between two independent
random sequences. We also obtain tail bounds on this quantity.
