Anti-lecture Hall Compositions and Overpartitions

William Y.C. Chen, Doris D.M. Sang and Diane Y.H. Shi

  Abstract:  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-Ramanjan 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.

  AMS Classification:  05A17, 11P84

  Keywords:  anti-lecture hall composition, Rogers-Ramanujan type identity, overpartition, Durfee dissection

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