An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

AbstractWe initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

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Lessons in Enumerative Combinatorics

10.1007/978-3-030-71250-1 ◽

2021 ◽

Author(s):

Ömer Eğecioğlu ◽

Adriano M. Garsia

Keyword(s):

Enumerative Combinatorics

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Combinatorics and Genus of Tropical Intersections and Ehrhart Theory

SIAM Journal on Discrete Mathematics ◽

10.1137/09075024x ◽

2010 ◽

Vol 24 (1) ◽

pp. 17-32 ◽

Cited By ~ 13

Author(s):

Reinhard Steffens ◽

Thorsten Theobald

Keyword(s):

Ehrhart Theory

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An Enumerative Combinatorics Model for Fragmentation Patterns in RNA Sequencing Provides Insights into Nonuniformity of the Expected Fragment Starting-Point and Coverage Profile

Journal of Computational Biology ◽

10.1089/cmb.2016.0096 ◽

2017 ◽

Vol 24 (3) ◽

pp. 200-212 ◽

Cited By ~ 1

Author(s):

Celine Prakash ◽

Arndt Von Haeseler

Keyword(s):

Rna Sequencing ◽

Enumerative Combinatorics ◽

Starting Point ◽

Fragmentation Patterns ◽

Coverage Profile

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Tabula III: Kepler’s Mysterious Polyhedral Model

Journal for the History of Astronomy ◽

10.1177/0021828617721545 ◽

2017 ◽

Vol 48 (3) ◽

pp. 281-311 ◽

Cited By ~ 1

Author(s):

Noam Andrews

Keyword(s):

Decorative Arts ◽

Major Work ◽

Polyhedral Model ◽

Polyhedral Geometry

The article addresses the genesis and visualization of the capstone image to Kepler’s polyhedral hypothesis of the planetary intervals from his first major work, Mysterium Cosmographicum (1596). The contention is that the famous Tabula III was directed less by Kepler than it was an initiative spearheaded by Georg Gruppenbach, the printer of Mysterium, and Kepler’s mentor Michael Mäistlin, who sought to produce a marketable broadsheet that would appeal to the contemporary German fashion for illustrations of polyhedral geometry. More generally, the article seeks to redefine the key role played by the printing workshop and the decorative arts in the theory’s inception and ultimate graphic manifestation.

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