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 $g$.