Abstract
For a non-decreasing integer sequence a=(a1,...,an) we define La to be the set of n-tuples of integers λ = (λ1,...,λn) 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 ai=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 Pθ,” for certain real numbers θ and integer sets Pθ. We then underline the connection between lecture hall partitions and Ehrhart theory and discuss some reciprocity results.
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References
G.E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Addison-Wesley, 1976.
G.E. Andrews, “MacMahon's partition analysis: I. The lecture hall partition theorem,” preprint, 1996.
M. Bousquet-Mélou and K. Eriksson, “Lecture hall partitions,” The Ramanujan Journal 1(1997), 101-111.
L. Euler, Introduction in Analysin Infinitorum, Chapter 16. Marcum-Michaelem Bousquet, Lausannae, 1748. French translation: Introduction à l'analyse Infinitésimale, ACL éditions, Paris, 1987.
P.A. MacMahon, Combinatory Analysis, 2 vols., Cambridge University Press, Cambridge, 1915-1916 (Reprinted: Chelsea, New-York, 1960).
R.P. Stanley, “Combinatorial reciprocity theorems,” Adv. Math. 14(1974), 194-253.
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Bousquet-Mélou, M., Eriksson, K. Lecture Hall Partitions II. The Ramanujan Journal 1, 165–185 (1997). https://doi.org/10.1023/A:1009768118404
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DOI: https://doi.org/10.1023/A:1009768118404