35 research outputs found
Unimodality Problems in Ehrhart Theory
Ehrhart theory is the study of sequences recording the number of integer
points in non-negative integral dilates of rational polytopes. For a given
lattice polytope, this sequence is encoded in a finite vector called the
Ehrhart -vector. Ehrhart -vectors have connections to many areas of
mathematics, including commutative algebra and enumerative combinatorics. In
this survey we discuss what is known about unimodality for Ehrhart
-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al.
(eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This
version updated October 2017 to correct an error in the original versio
The Lecture Hall Parallelepiped
The s-lecture hall polytopes P_s are a class of integer polytopes defined by
Savage and Schuster which are closely related to the lecture hall partitions of
Eriksson and Bousquet-M\'elou. We define a half-open parallelopiped Par_s
associated with P_s and give a simple description of its integer points. We use
this description to recover earlier results of Savage et al. on the
\delta-vector (or h^*-vector) and to obtain the connections to s-ascents and
s-descents, as well as some generalizations of these results.Comment: 14 pages. To appear in Annals of Combinatoric
Polyhedral geometry for lecture hall partitions
Lecture hall partitions are a fundamental combinatorial structure which have
been studied extensively over the past two decades. These objects have produced
new results, as well as reinterpretations and generalizations of classicial
results, which are of interest in combinatorial number theory, enumerative
combinatorics, and convex geometry. In a recent survey of Savage
\cite{Savage-LHP-Survey}, a wide variety of these results are nicely presented.
However, since the publication of this survey, there have been many new
developments related to the polyhedral geometry and Ehrhart theory arising from
lecture hall partitions. Subsequently, in this survey article, we focus
exclusively on the polyhedral geometric results in the theory of lecture hall
partitions in an effort to showcase these new developments. In particular, we
highlight results on lecture hall cones, lecture hall simplices, and lecture
hall order polytopes. We conclude with an extensive list of open problems and
conjectures in this area.Comment: 20 pages; To appear in to proceedings of the 2018 Summer Workshop on
Lattice Polytopes at Osaka Universit