Lecture Hall Partitions II

For a non-decreasing integer sequence a=(a₁,...,a_n) we define L_a to be the set of n-tuples of integers = (₁,...,_n) satisfying . This generalizes the so-called lecture hall partitions corresponding to a_i=i and previously studied by the authors and by Andrews. We find sequences a such that the weight generating function for these a-lecture hall partitions has the remarkable form In the limit when n tends to infinity, we obtain a family of identities of the kind the number of partitions of an integer m such that the quotient between consecutive parts is greater than is equal to the number of partitions of m into parts belonging to the set P, for certain real numbers and integer sets P. We then underline the connection between lecture hall partitions and Ehrhart theory and discuss some reciprocity results.