An important problem in mathematics is understanding the behavior of functions. If we can show that complicated functions can be simplified, computation becomes more feasible. In this thesis, we describe two major projects: counting arcs in projective space and counting subrings in $\Z^n$. In the case of arcs in $\mb{P}^{k-1}(\Fq)$, we seek to understand whether the counting function can be expressed by a simple formula like a polynomial. In the case of subrings of $\Z^n$, we are interested in the asymptotic growth of the function that counts subrings of index at most $X$.

An $n$-arc in $(k-1)$-dimensional projective space is a set of $n$ points so that no $k$ lie on a hyperplane. Let $C_{n,k}(q)$ be the number of $n$-arcs in $\mb{P}^{k-1}(\Fq)$. We discuss new results for $n$-arcs when $k = 3$ and 4. Building off work of Kaplan, Kimport, Lawrence, Peilen, and Weinreich, we show that $C_{10,3}(q)$ is not quasipolynomial in $q$. For almost all $n$, no formulas for $C_{n,k}(q)$ were known when $k \ge 4$. We introduce a new algorithm to count $n$-arcs in $\mb{P}^3(\Fq)$ in terms of a small number of special combinatorial objects and use it to compute $C_{n,4}(q)$ for $n \le 7$. Finally, we discuss generalizations of this algorithm to higher-dimensional projective space.

Let $f_n(p^e)$ count the number of subrings of index $p^e$ in $\Z^n$. We study lower bounds for $f_n(p^e)$ using techniques from combinatorics and arithmetic geometry. Using these lower bounds, we give the best known result about the asymptotic growth of subrings in $\Z^n$ when $n \ge 8$ by studying the analytic properties of the subring zeta function for $\Z^n$. These results immediately give new information about the asymptotic growth of orders in a fixed degree $n$ number field. Finally, we consider the behavior of $f_n(p^e)$ as a function of $p$. We give evidence that this function may always be polynomial in $p$.

Several recent papers have explored families of rational polyhedra whose integer points are in bijection with certain families of numerical semigroups. One such family, first introduced by Kunz, has integer points in bijection with numerical semigroups of fixed multiplicity, and another, introduced by Hellus and Waldi, has integer points corresponding to oversemigroups of numerical semigroups with two generators. In this paper, we provide a combinatorial framework from which to study both families of polyhedra. We introduce a new family of polyhedra called group cones, each constructed from some finite abelian group, from which both of the aforementioned families of polyhedra are directly determined but that are more natural to study from a standpoint of polyhedral geometry. We prove that the faces of group cones are naturally indexed by a family of finite posets, and illustrate how this combinatorial data relates to semigroups living in the corresponding faces of the other two families of polyhedra.

Keywords: Polyhedron, numerical semigroup.

Mathematics Subject Classifications: 52B05, 20M14

A partition is an $a$-core partition if none of its hook lengths are divisible by $a$. It is well known that the number of $a$-core partitions is infinite and the number of simultaneous $(a, b)$-core partitions is a generalized Catalan number if $a$ and $b$ are relatively prime. In the first half of the dissertation, we give an expression for the number of simultaneous $(a_1,a_2,\dots, a_k)$-core partitions that is equal to the number of integer points in a polytope.

In the second half, we discuss objects closely related to core partitions, called numerical semigroups, which are additive monoids that have finite complements in the set of non-negative integers. For a numerical semigroup $S$, the genus of $S$ is the number of elements in $\NN \setminus S$ and the multiplicity is the smallest nonzero element in $S$. In 2008, Bras-Amor os conjectured that the number of numerical semigroups with genus $g$ is increasing as $g$ increases. Later, Kaplan posed a conjecture that implies Bras-Amor os conjecture. In this dissertation, we prove Kaplan's conjecture when the multiplicity is 4 or 6 by counting the number of integer points in a polytope. Moreover, we find a formula for the number of numerical semigroups with multiplicity 4 and genus $g$.

Let $E/\QQ$ be an elliptic curve; for all but finitely many primes $p$, reduction modulo $p$ yields an elliptic curve over the finite field $\mathbb{F}_p$, and it is natural to ask about the properties of these reductions for varying primes. The purpose of this dissertation is to study one such question, namely, how frequently the reductions result in an elliptic curve with cyclic group structure. To be precise, we let $\pi_E^{cyc}(x) $ denote the number of primes less than $x$ for which the reduction of $E$ modulo $p$ is cyclic. The asymptotic behavior of this function has been established by Serre conditional on Generalized Riemann Hypothesis. Furthermore, Banks and Shparlinski showed that this asymptotic holds unconditionally on average over the family of elliptic curves given by short Weierstrass equations with coefficients taken in a `box.' Inspired by the work of Battista, Bayless, Ivanov and James on the Lang-Trotter conjecture, we study the average asymptotic behavior of the functions $\pi_E^{cyc}$ where the average is taken over certain thin families of elliptic curves: elliptic curves with a rational point of order $m$ defined over $\mathbb{Q}$. The results we obtain are again in agreement with the conditional asymptotic. We also extend the study of cyclicity from elliptic curves defined over the rational numbers to elliptic curves defined over a quadratic extension of $\QQ$ and obtain partial results in that case.As a key tool, we prove an analogue of a result of Vl\u{a}du\c{t} that estimates the number of elliptic curves over a finite field which have some specified torsion and which have group structure that is as cyclic as possible.

Our results are concerned with couplings, component counts of combinatorial objects, and

probabilistic number theory. In the theory of couplings, we are concerned with the general

problem of proving the existence of joint distributions p(i,j) of two discrete random variables M and

N subject to infinitely many constraints of the form p(M=i, N=j)=0. The constraints placed on the joint distributions will require, for many

elements j in the range of N, p(M=i, N=j)=0 for infinitely many values of i in the

range of M, where the corresponding values of i depend on j. To prove the existence of

such joint distributions, we apply a theorem proved by Volker Strassen on the existence of joint

distributions with prespecified marginal distributions. In the case in which N is uniformly

distributed in a combinatorial structure with C_i components of size i, we seek to measure

the amount of dependence in the process (C_i )_{i\le n} by coupling N with a variable M such that

M has Z_i components of size i and the Z_i's are independent, with

sum_{i\le n}(C_i - Z_i)^+ <2.

In the combinatorial example of noncrossing partitions, we provide

two derivations of the probability distribution of the component counts of a uniformly distributed noncrossing partition. Upon applying a bijection between the set of noncrossing partitions

and Dyck paths consisting of up-steps and down-steps, our results specify the joint and marginal distributions of the block counts of the number of consecutive up-steps in a uniformly random chosen Dyck path.

In number theory, we give an analogue of the Erd\"{o}s-Kac

Theorem by providing a family of integer-valued random variables on

{1,\ldots,n} whose number of distinct prime factors

is roughly log\log n+X\cdot\sqrt{\log\log n} for large values

of n, where X is a standard normal variable. Our final result

involves couplings of a Zeta-distributed variable. Given s>1 and

n\in\mathbb{N}, consider a Zeta(s)-distributed integer-valued

random variable with prime factorization Z(s)=\prod_{p}p^{\alpha_{p}(s)}

and the truncation Z_{n}(s)\coloneqq\prod_{p\le n}p^{\alpha_{p}(s)

The prime powers \alpha_{p}(s) are independent with

\alpha_{p}(s)\sim\text{Geometric}(1/p^s),

and we also consider a random variable M(n)=\prod_{p\le n}p^{Z_{p}},

where the Z_{p}'s are independent with Z_{p}\sim\text{Geometric}(1/p).

We apply the concept of pivot mass and a theorem proved by Strassen

in order to prove the existence of couplings of a Zeta-distributed random variable Z(s) and M(n)

in which we can make probabilistic divisibility statements of the form "Z_{n}(s) divides M(n)P(n)"

for some random prime P(n)\le n. In particular, we will

prove that for each n\in\mathbb{N} and an integer k\ge 4, there exists an \varepsilon(k)>0

such that when s\in\left(1,1+\varepsilon(k) we can

couple Z(s) and M\left(n\right) such that if Z(s)\le k,

then Z_{n}(s) always divides M(n)P(n)

for some random prime P(n)\le n.