Liam Solus

Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices

A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart)

r-Stable hypersimplices

h^\ast

Borromean rays and hyperplanes

-polynomial. This conjecture can be viewed as a strengthening of a previously disproved conjecture which stated that any Gorenstein lattice polytope has a unimodal

Counting Markov equivalence classes for DAG models on trees

h^\ast

Polyhedral Problems in Combinatorial Convex Geometry

-polynomial. The first counterexamples to unimodality for Gorenstein lattice polytopes were given in even dimensions greater than five by Musta{ţ}{ǎ} and Payne, and this was extended to all dimensions greater than five by Payne. While there exist numerous examples in support of the conjecture that IDP reflexives are

A Shelling of the Odd Second Hypersimplex

h^\ast

Consistency Guarantees for Permutation-Based Causal Inference Algorithms

-unimodal, its validity has not yet been considered for families of reflexive lattice simplices that closely generalize Paynes counterexamples. The main purpose of this work is to prove that the former conjecture does indeed hold for a natural generalization of Paynes examples. The second purpose of this work is to extend this investigation to a broader class of lattice simplices, for which we present new results and open problems.

Permutation-based Causal Inference Algorithms with Interventions

Abstract Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called r-stable hypersimplices, and show that a well-known regular unimodular triangulation of the hypersimplex restricts to a triangulation of each r-stable hypersimplex. For the case of the second hypersimplex defined by the two-element subsets of an n-set, we provide a shelling of this triangulation that sequentially shells each r-stable sub-hypersimplex. In this case, we utilize the shelling to compute the Ehrhart h ⁎ -polynomials of these polytopes, and the hypersimplex, via independence polynomials of graphs. For one such r-stable hypersimplex, this computation yields a connection to CR mappings of Lens spaces via Ehrhart–MacDonald reciprocity.

Shellability, Ehrhart Theory, and

Three disjoint rays in euclidean 3-space form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.

r

Abstract DAG models are statistical models satisfying a collection of conditional independence relations encoded by the nonedges of a directed acyclic graph (DAG) G . Such models are used to model complex cause–effect systems across a variety of research fields. From observational data alone, a DAG model G is only recoverable up to Markov equivalence. Combinatorially, two DAGs are Markov equivalent if and only if they have the same underlying undirected graph (i.e., skeleton) and the same set of the induced subDAGs i → j ← k , known as immoralities. Hence it is of interest to study the number and size of Markov equivalence classes (MECs). In a recent paper, we introduced a pair of generating functions that enumerate the number of MECs on a fixed skeleton by number of immoralities and by class size, and we studied the complexity of computing these functions. In this paper, we lay the foundation for studying these generating functions by analyzing their structure for trees and other closely related graphs. We describe these polynomials for some well-studied families of graphs including paths, stars, cycles, spider graphs, caterpillars, and balanced binary trees. In doing so, we recover connections to independence polynomials, and extend some classical identities that hold for Fibonacci numbers. We also provide tight lower and upper bounds for the number and size of MECs on any tree. Finally, we use computational methods to show that the number and distribution of high degree nodes in a triangle-free graph dictate the number and size of MECs.