Generating functions and triangulations for lecture hall cones

Matthias Beck; Benjamin Braun; Matthias Köppe; Carla D. Savage; Zafeirakis Zafeirakopoulos

doi:10.1137/15M1036907

Generating functions and triangulations for lecture hall cones

Matthias Beck, Benjamin Braun, Matthias Köppe, Carla D. Savage, Zafeirakis Zafeirakopoulos

Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

We investigate the arithmetic-geometric structure of the lecture hall cone L_n := {∈ ℝⁿ : 0 ≤ 1/1 ≤ 2/1 ≤ 3/1 ≤ ...≤ n/n}. We show that Ln is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart h^∗-polynomial is given by the (n-1)st Eulerian polynomial and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for L_n, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of L_n, including connections between enumerative and algebraic properties of Ln and cones over unit cubes.

Original language	English
Pages (from-to)	1470-1479
Number of pages	10
Journal	SIAM Journal on Discrete Mathematics
Volume	30
Issue number	3
DOIs	https://doi.org/10.1137/15M1036907
State	Published - 2016

Bibliographical note

Publisher Copyright:
Copyright © by SIAM.

Funding

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	1162638, 0914873

Keywords

Eulerian
Generating functions
Lecture hall
Triangulations

ASJC Scopus subject areas

General Mathematics

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10.1137/15M1036907

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title = "Generating functions and triangulations for lecture hall cones",

abstract = "We investigate the arithmetic-geometric structure of the lecture hall cone Ln := \{∈ ℝn : 0 ≤ 1/1 ≤ 2/1 ≤ 3/1 ≤ ...≤ n/n\}. We show that Ln is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart h∗-polynomial is given by the (n-1)st Eulerian polynomial and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for Ln, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of Ln, including connections between enumerative and algebraic properties of Ln and cones over unit cubes.",

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language = "English",

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T1 - Generating functions and triangulations for lecture hall cones

AU - Beck, Matthias

AU - Braun, Benjamin

AU - Köppe, Matthias

AU - Savage, Carla D.

AU - Zafeirakopoulos, Zafeirakis

PY - 2016

Y1 - 2016

N2 - We investigate the arithmetic-geometric structure of the lecture hall cone Ln := {∈ ℝn : 0 ≤ 1/1 ≤ 2/1 ≤ 3/1 ≤ ...≤ n/n}. We show that Ln is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart h∗-polynomial is given by the (n-1)st Eulerian polynomial and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for Ln, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of Ln, including connections between enumerative and algebraic properties of Ln and cones over unit cubes.

AB - We investigate the arithmetic-geometric structure of the lecture hall cone Ln := {∈ ℝn : 0 ≤ 1/1 ≤ 2/1 ≤ 3/1 ≤ ...≤ n/n}. We show that Ln is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart h∗-polynomial is given by the (n-1)st Eulerian polynomial and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for Ln, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of Ln, including connections between enumerative and algebraic properties of Ln and cones over unit cubes.

KW - Eulerian

KW - Generating functions

KW - Lecture hall

KW - Triangulations

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DO - 10.1137/15M1036907

M3 - Article

AN - SCOPUS:84990960808

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VL - 30

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JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 3

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Generating functions and triangulations for lecture hall cones

Abstract

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Keywords

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