Anti-lecture hall compositions and overpartitions

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,±1 modulo k. This identity can be considered as a finite version of the anti-lecture hall theorem of Corteel and Savage. To prove this result, we find two Rogers–Ramanujan type identities for overpartitions which are analogous to the Rogers–Ramanujan type identities due to Andrews. When k is odd, we give another proof by using the bijections of Corteel and Savage for the anti-lecture hall theorem and the generalized Rogers–Ramanujan identity also due to Andrews.