For b=(b1,⋯,bn)∈Z>0n, a b-parking function is defined to be a sequence (β1,⋯,βn) of positive integers whose nondecreasing rearrangement β1′≤β2′≤⋯≤βn′ satisfies βi′≤b1+⋯+bi. The b-parking-function polytope Xn(b) is the convex hull of all b-parking functions of length n in Rn. Geometric properties of Xn(b) were previously explored in the specific case where b=(a,b,b,⋯,b) and were shown to generalize those of the classical parking-function polytope. In this work, we study Xn(b) in full generality. We present a minimal inequality and vertex description for Xn(b), prove it is a generalized permutahedron, and study its h-polynomial. Furthermore, we investigate Xn(b) through the perspectives of building sets and polymatroids, allowing us to identify its combinatorial types and obtain bounds on its combinatorial and circuit diameters.

This thesis is comprised of eight articles in the fields of algebraic, geometric and enumerative combinatorics, as well as algebraic statistics and causality. These works are motivated by problems in the mentioned areas, and have polytopes as their underlying object of study. The investigated properties of these polytopes include the distribu-tional properties of their associated combinatorial generating polynomials, lattice point enumeration, face structure, and properties of their corresponding toric ideals. These investigations, for instance, provide answers to some open questions in combinatorics as well as ne wmethodologies for causal discovery. The main characters are lattice polytopes, simplicial complexes, generating functions, permutations,graphs, posets, and statistical models. These objects often interactin rich and surprising ways. 

The Ehrhart polynomial ehr(P)(n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f*-vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of ehr(P)(n) with respect to the binomial coefficient basis {((n-1)(0)),((n-1)(1)),& mldr;,((n-1)(d))}, where d = dim P. Similarly to h/h*-vectors, the f*-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f*-vectors of lattice polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f*-coefficients increases and the last quarter decreases. Even though f*-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h*-vector, there is a polytope with the same h*-vector whose f*-vector is unimodal.

We consider the problem of estimating the marginal independence structure of a Bayesian network from observational data, learning an undirected graph we call the unconditional dependence graph. We show that unconditional dependence graphs of Bayesian networks correspond to the graphs having equal independence and intersection numbers. Using this observation, a Gröbner basis for a toric ideal associated to unconditional dependence graphs of Bayesian networks is given and then extended by additional binomial relations to connect the space of all such graphs. An MCMC method, called GrUES (Gröbner-based unconditional equivalence search), is implemented based on the resulting moves and applied to synthetic Gaussian data. GrUES recovers the true marginal independence structure via a penalized maximum likelihood or MAP estimate at a higher rate than simple independence tests while also yielding an estimate of the posterior, for which the 20% HPD credible sets include the true structure at a high rate for data-generating graphs with density at least 0.5.

The Ehrhart polynomial ehrP(n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f∗-vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of ehrP(n) with respect to the binomial coefficient basis (Formula presented.), where d = dimP. Similarly to h/h∗-vectors, the f∗-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f∗-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f∗-coefficients increases and the last quarter decreases. Even though f∗-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h∗-vector, there is a polytope with the same h∗-vector whose f∗-vector is unimodal.

We consider the problem of characterizing Bayesian networks up to unconditional equivalence, i.e., when directed acyclic graphs (DAGs) have the same set of unconditional $d$-separation statements. Each unconditional equivalence class (UEC) is uniquely represented with an undirected graph whose clique structure encodes the members of the class. Via this structure, we provide a transformational characterization of unconditional equivalence; i.e., we show that two DAGs are in the same UEC if and only if one can be transformed into the other via a finite sequence of specified moves. We also extend this characterization to the essential graphs representing the Markov equivalence classes (MECs) in the UEC. UECs form a partition coarsening of the space of MECs and are easily estimable from marginal independence tests. Thus, a characterization of unconditional equivalence has applications in methods that involve searching the space of MECs of Bayesian networks.

For a finite group G, denote by α(G) the minimum number of vertices of any graph having Aut() ∼= G. In this paper, we prove that α(G) ≤ |G|, with specifiedexceptions. The exceptions include four infinite families of groups, and 17 other smallgroups. Additionally, we compute α(G) for the groups G such that α(G) > |G| wherethe value α(G) was previously unknown.