Lascoux, Leclerc, and Thibon defined a remarkable family of symmetric functions that are q-deformations of products of skew Schur functions. These LLT polynomials G_\lambda(x;q) can be indexed by a tuple \lambda of skew diagrams. When each skew diagram is a row, we define a weighted graph \Pi(\lambda). We show that a horizontal-strip LLT polynomial is determined by this weighted graph. When \Pi(\lambda) has no triangles, we establish a combinatorial Schur expansion of G_\lambda(x;q). We also explore a connection to extended chromatic symmetric functions.

LLT polynomials were first introduced by Lascoux, Leclerc, and Thibon using the action of an affine Hecke algebra for S$_n$, and can be viewed as a $q$-generating function for both ribbon tableaux and tuples of semistandard Young tableaux. This definition has since been expanded to arbitrary Lie type, although with no combinatorial definition. We establish a combinatorial model for LLT polynomials in particular cases for Sp$_{2n}$ and further conjecture a similar model for the orthogonal Lie types. Our definition uses a new object we call an out-in tableau as well as a correspondence between oscillating tableaux and symplectic tableaux that we use to give a proof of a Cauchy identity for Sp$_{2n}$.

We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, including Bayer's conjecture on equations defining Grothendieck's classical Hilbert scheme and the construction of a Chow morphism for toric Hilbert schemes.

This thesis proves a special case of the $k$-Littlewood--Richardson rule, which is analogous to the classical Littlewood--Richardson rule but is used in the case for $k$-Schur functions. The classical Littlewood--Richardson rule gives a combinatorial formula for the coefficients $c_{\mu\nu}^{\lambda}$ appearing in the expression $s_{\mu}s_{\nu} = \sum_{\lambda} c_{\mu\nu}^{\lambda}s_{\lambda}$ for two Schur functions multiplied together. $k$-Schur functions are another class of symmetric functions which were introduced by Lapointe, Lascoux, and Morse and are indexed by and related to $k$-bounded partitions. We investigate what occurs when multiplying two $k$-Schur functions with some restrictions. More specifically, we investigate what happens when a $k$-Schur function is multiplied by a $k$-Schur function corresponding to a partition of length two. In this restricted case we are able to provide a combinatorial description of the $k$-Littlewood--Richardson coefficients that appear in the expansion of the product as a sum of $k$-Schur functions. These $k$-Littlewood--Richardson coefficients can be computed in terms of the number of $k$-tableaux with a certain property we call $k$-lattice. Furthermore, we conjecture that the result holds for any $k$-Schur functions, even when no restrictions are imposed. The proofs presented rely on a class of operators on $k$-tableaux which we introduce that are similar to the crystal operators on classical tableaux, but we provide a specific example that implies they are not actually crystal operators on $k$-tableaux. In addition to this, we also provide numerous examples and dedicate a chapter to examples of computation for some $k$-Littlewood--Richardson coefficients.

In geometric, algebraic, and topological combinatorics, properties such as symmetry, unimodality, and real-rootedness of combinatorial generating polynomials are frequently studied. In this thesis, we present three projects exploring various properties of polynomials arising from Ehrhart theory, the study of counting integer points in lattice polytopes. Many of the open questions on real-rootedness and unimodality of polynomials pertain to the enumeration of faces of cell complexes. When proving that a polynomial is real-rooted, we often rely on the theory of interlacing polynomials and their recursive nature. We relate the theory of interlacing polynomials to the shellability of cell complexes. We first derive a sufficient condition for stability of the $h$-polynomial of a subdivision of a shellable complex. To apply it, we generalize the notion of reciprocal domains for convex embeddings of polytopes to abstract polytopes and use this generalization to define the family of stable shellings of a polytopal complex. We characterize the stable shellings of cubical and simplicial complexes, and apply this theory to answer a question of Brenti and Welker on barycentric subdivisions for the well-known cubical polytopes. We also give a positive solution to a problem of Mohammadi and Welker on edgewise subdivisions of cell complexes. We end by relating the family of stable line shellings to the combinatorics of hyperplane arrangements.

The Ehrhart polynomial $\text{ehr}_P(t)$ 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 $\text{ehr}_P(n)$ with respect to the binomial coefficient basis $ \left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}$, where $d = \text{dim} P$. 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 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.

Posets can be viewed as subsets of the type-A root system that satisfy certain properties. Geometric objects arising from posets, such as order cones, order polytopes, and chain polytopes, have been widely studied, though many open questions concerning them remain open, such as the unimodality of their $h^*$-vectors. In 1993, Vic Reiner introduced signed posets, which are subsets of the type-B root system that satisfy the same properties. We introduce the analogue of order and chain polytopes in this setting, focusing on the Ehrhart theory of these objects. We are able to determine when these signed order polytopes have symmetric $h^*$-vectors, and end with a discussion of open questions regarding signed chain polytopes.

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation H_mu(X;q,t)=H_{mu*}(X;t,q). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q=0) when mu is a partition with at most three rows, and for the coefficients of the square-free monomials in X={x_1,x_2,...} for all shapes mu. We also provide a proof for the full relation in the case when mu is a hook shape, and for all shapes at the specialization t=1. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.

We investigate three graph polynomials including antimagic, super edge-magic, and chromatic polynomials. The Antimagic Graph Conjecture asserts that every connected graph $G = (V,E)$ except $K_2$ admits an edge labeling such that each label $1,2,\ldots,|E|$ is used exactly once and the sums of the labels of the edges incident to each vertex are distinct. We introduce partially magic labelings where the vertex sums are the same in a subset of $V$. By using the quasi-polynomial structure of the partially magic labeling counting function, we show that every bipartite graph satisfies a relaxed version of the Antimagic Graph Conjecture (that is, repetition of labels are allowed). \

A total labeling $f:E \cup V \to \ZZ_{\ge 0}$ of a graph $G=(V,E)$ is called \emph{super edge-magic} if each vertex label is in $\{1,\ldots, |V|\}$ and the sum of the edge label plus labels of its two ends is the same for all edges of $G$. We prove that the counting function of super edge-magic labelings of every tree is a polynomial. This helps us to show that every tree admits a relaxed super edge magic labeling which is the relaxed version of \emph{Super Edge-Magic Tree Conjecture}. Moreover, we show that every tree with one extra edge that makes a unique even cycle admits a super edge-magic labeling. On the other hand, a \emph{harmonious labeling} is an injective function $L:V \rightarrow \{0, 1, \ldots, |E|-1 \}$ such that the induced edge labels $L^{*}(e) \equiv L(u)+L(v)$ (mod $|E|$) for every edge $e=\{u,v\} \in E,$ are distinct. The \emph{Harmonious Tree Conjecture} indicates that every tree admits a harmonious labeling. We use our results on super edge-magic labelings to prove that every tree admits a relaxed version of the Harmonious Tree Conjecture. \

Lastly, we extend classic order polynomials to a two variable version and we drive a reciprocity theorem for strict and weak order polynomials, reminiscent of Stanley's reciprocity theorem. We also study the \emph{bivariate chromatic polynomial} which counts the number of vertex coloring of $G$ with $1 \le x$ colors allowing the same colors for adjacent vertices if the color is $\ge y$ ($1 \le y \le x$). We decompose bivariate chromatic polynomial into bivariate order polynomials and we find a reciprocity relation linking bivariate chromatic polynomials to acyclic orientations.

Both LLT polynomials and k-Schur functions were derived from the study of Macdonald polynomials, and have proved to be fruitful areas of study. A well-known conjecture due to Haglund and Haiman states that k-bandwidth LLT polynomials expand positively into $k$-Schur functions. This is trivial in the case k=1 and has been recently proved for k=2. In this work, we present a proof for the case $k=3$. In doing so, we introduce a new method for establishing linear relations among LLT polynomials.

Abstract: We prove a combinatorial formula for the Macdonald polynomial $\tilde{H}_{\mu }(x;q,t)$ which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of $\tilde{H}_{\mu }(x;q,t)$ in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schützenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahi's combinatorial formula for Jack polynomials as well as a lifting of their formula to integral form Macdonald polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients $\tilde{K}_{\lambda \mu }(q,t)$ in the case that $\mu $ is a partition with parts $\leq 2$.

In 2001, Lapointe, Lascoux, and Morse discovered a class of symmetric functions called k-Schur functions. These functions have many properties similar to Schur functions and were motivated by a conjectured refinement of the Macdonald positivity conjecture. We describe a representation-theoretic model for k-Schur functions by studying the combinatorics of special pairs of partitions called skew-linked partitions. En route we also study nonnegative integer matrices with specified row and column sums. These data allow us to construct "small" modules that are generalizations of

Garsia-Procesi modules. We describe properties of k-Schur functions that can be deduced from these modules.