Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences
Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences
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DOI：
10.1016/j.jcta.2011.12.005
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发表时间：
2012-05
期刊：
影响因子：
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通讯作者：
C. Savage;Michael J. Schuster
中科院分区：
文献类型：
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作者：
C. Savage;Michael J. Schuster
对于正整数序列s=（s1，.，sn），s-讲堂划分是满足0 λ n/sn的整数序列λ。在这项工作中，我们介绍了s-演讲厅多面体，s-反转序列，和相关的统计两个家庭。我们证明了对任意正整数序列s：（i）s-讲座厅多面体的h-向量是相应的s-反演序列的上升多项式;（ii）s-反演序列的上升多项式推广了欧拉多项式，包括跟踪s-反演序列主指数推广的q-模拟;以及（iii）s-演讲厅分区的生成函数可以用s-欧拉多项式的新的q-模拟来解释，其跟踪关于s-反转序列的“演讲厅”统计。我们展示了如何通过三个S-家庭的分区，多面体和反转序列四个不同的统计相关。我们的方法使用Ehrhart理论，涉及的分区理论的演讲厅分区，他们的几何形状。
For a sequence s=(s1,…,sn) of positive integers, an s-lecture hall partition is an integer sequence λ satisfying 0⩽λ1/s1⩽λ2/s2⩽⋯⩽λn/sn. In this work, we introduce s-lecture hall polytopes, s-inversion sequences, and relevant statistics on both families. We show that for any sequence s of positive integers: (i) the h⁎-vector of the s-lecture hall polytope is the ascent polynomial for the associated s-inversion sequences; (ii) the ascent polynomials for s-inversion sequences generalize the Eulerian polynomials, including a q-analog that tracks a generalization of major index on s-inversion sequences; and (iii) the generating function for the s-lecture hall partitions can be interpreted in terms of a new q-analog of the s-Eulerian polynomials, which tracks a “lecture hall” statistic on s-inversion sequences. We show how four different statistics are related through the three s-families of partitions, polytopes, and inversion sequences. Our approach uses Ehrhart theory to relate the partition theory of lecture hall partitions to their geometry.