  We show that the number of anti-lecture hall compositions of n with the first
entry not exceeding k-2 equals the number of overpartitions of n with
non-overlined parts not congruent to $0,\pm 1$ modulo k. This identity can be
considered as a refined version of the anti-lecture hall theorem of Corteel and
Savage. To prove this result, we find two Rogers-Ramanujan type identities for
overpartition which are analogous to the Rogers-Ramanjan type identities due to
Andrews. When k is odd, we give an alternative proof by using a generalized
Rogers-Ramanujan identity due to Andrews, a bijection of Corteel and Savage and
a refined version of a bijection also due to Corteel and Savage.
