8 citations · 23 across the 15 of their papers we have counts for
20 papers
Weighted Ehrhart Theory: Extending Stanley's nonnegativity theorem
Esme Bajo, Robert Davis, Jesús A. De Loera +4
We generalize R. P. Stanley's celebrated theorem that the -polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under contai…
Perfectly Matchable Set Polynomials and -polynomials for Stable Set Polytopes of Complements of Graphs
Robert Davis, Florian Kohl
A subset of vertices of a graph is called a perfectly matchable set of if the subgraph induced by contains a perfect matching. The perfectly matchable set polynomia…
Normalized Volumes of Type-PQ Adjacency Polytopes for Certain Classes of Graphs
Robert Davis, Joakim Jakovleski, Qizhe Pan
The type-PQ adjacency polytope associated to a simple graph is a -polytope containing valuable information about an underlying power network. Chen and the first author have re…
Facets and facet subgraphs of symmetric edge polytopes
Tianran Chen, Robert Davis, Evgeniia Korchevskaia
Symmetric edge polytopes, a.k.a. PV-type adjacency polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including numbe…
The Integer Decomposition Property and Weighted Projective Space Simplices
Benjamin Braun, Robert Davis, Derek Hanely +2
Reflexive lattice polytopes play a key role in combinatorics, algebraic geometry, physics, and other areas. One important class of lattice polytopes are lattice simplices defining…
Computing Volumes of Adjacency Polytopes via Draconian Sequences
Robert Davis, Tianran Chen
Adjacency polytopes appear naturally in the study of nonlinear emergent phenomena in complex networks. The "PQ-type" adjacency polytope, denoted and which…
Universal Gröbner Bases of Toric Ideals of Combinatorial Neural Codes
Melissa Beer, Robert Davis, Thomas Elgin +5
In the 1970s, O'Keefe and Dostrovsky discovered that certain neurons, called place cells, in an animal's brain are tied to its location within its arena. A combinatorial neural cod…
A toric deformation method for solving Kuramoto equations
Tianran Chen, Robert Davis
The study of frequency synchronization configurations in Kuramoto models is a ubiquitous mathematical problem that has found applications in many seemingly independent fields. In t…
State Polytopes Related to Two Classes of Combinatorial Neural Codes
Robert Davis
Combinatorial neural codes are vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine…
A Polyhedral Proof of a Wreath Product Identity
Robert Davis, Bruce Sagan
In 2013, Beck and Braun proved and generalized multiple identities involving permutation statistics via discrete geometry. Namely, they recognized the identities as specializations…
A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes
Tianran Chen, Robert Davis
The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In…
Counting equilibria of the Kuramoto model using birationally invariant intersection index
Tianran Chen, Robert Davis, Dhagash Mehta
Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amo…
Descents and des-Wilf Equivalence of Permutations Avoiding Certain Non-Classical Patterns
Caden Bielawa, Robert Davis, Daniel Greeson +1
A frequent topic in the study of pattern avoidance is identifying when two sets of patterns are Wilf equivalent, that is, when for al…
The pinnacle set of a permutation
Robert Davis, Sarah A. Nelson, T. Kyle Petersen +1
The peak set of a permutation records the indices of its peaks. These sets have been studied in a variety of contexts, including recent work by Billey, Burdzy, and Sagan, which enu…
Width- Generalizations of Classical Permutation Statistics
Robert Davis
We introduce new natural generalizations of the classical descent and inversion statistics for permutations, called width- descents and width- inversions. These variations in…
Pattern-Avoiding Polytopes
Robert Davis, Bruce Sagan
Two well-known polytopes whose vertices are indexed by permutations in the symmetric group are the permutohedron and the Birkhoff polytope . We consider…
Detecting the Integer Decomposition Property and Ehrhart Unimodality in Reflexive Simplices
Benjamin Braun, Robert Davis, Liam Solus
A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) -polynomi…
Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices
Robert Davis
In Ehrhart theory, the -vector of a rational polytope often provide insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polyto…
Ehrhart series, unimodality, and integrally closed reflexive polytopes
Benjamin Braun, Robert Davis
An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal -vector. Although various sufficient conditions have been found, necessa…
Compositions constrained by graph Laplacian minors
Benjamin Braun, Robert Davis, Ashley Harrison +3
Motivated by examples of symmetrically constrained compositions, super convex partitions, and super convex compositions, we initiate the study of partitions and compositions constr…