Lecture Hall Partitions II

Abstract

For a non-decreasing integer sequence a=(a₁,...,a_n) we define L_a to be the set of n-tuples of integers λ = (λ₁,...,λ_n) satisfying $0 \leqslant \frac{{\lambda _{\text{1}} }}{{a_1 }} \leqslant \frac{{\lambda _2 }}{{a_2 }} \leqslant \cdots \leqslant \frac{{\lambda _n }}{{a_n }}$. 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 $\sum\limits_{\lambda \in L_a } {q^{|\lambda |} } = \frac{1}{{(1 - q^{e_1 } )(1 - q^{e_2 } ) \cdot \cdot \cdot (1 - q^{e_n } )}}$

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 θ is equal to the number of partitions of m into parts belonging to the set P_θ,” for certain real numbers θ and integer sets P_θ. We then underline the connection between lecture hall partitions and Ehrhart theory and discuss some reciprocity results.