Some of the most interesting and fundamental objects in the study of combinatorics are permutations. {\em Permutations} are typically defined to be an arrangement of the numbers $\{1,2, \ldots n\}$, and they appear in countless problems and applications throughout mathematics. Sometimes we are particularly interested in observing or counting some key property of a given permutation. A {\em permutation statistic} is a function that takes a particular permutation and returns specific information about it, such as how many numbers are only adjacent to smaller numbers. Each permutation statistic gives rise to many counting questions, such as finding all possible permutations which have that particular statistic, or perhaps finding a subset of those permutations where that statistic is maximized. These numbers can sometimes be described by generating functions or polynomials which give light to the beauty and structure of mathematics, sometimes even in fields outside combinatorics. Some examples of this will be given in Chapter 1. In the rest of this dissertation, we will explore several of these statistics including new ones and variations on some of the more well known statistics. In the Chapter 2, we focus on shuffle compatibility for cyclic permutations. The {\em shuffle} of two permutations is the set of all permutations that have both original permutations as sub-sequences. A statistic is said to be {\em shuffle compatible} if its values over all possible shuffles of two permutations is completely determined by the statistic on the two original permutations, together with their lengths. Shuffle compatibility is implicit in Stanley's work on $P$-partitions and was also studied by Gessel and Zhuang. Shuffle compatibility is also useful in studying mathematical objects outside of combinatorics, such as quasisymmetric functions. More recently, an analogous definition of shuffle compatibility has been defined for {\em cyclic} permutations, which are permutations arranged in a circle so that the last element is considered adjacent to the first. In Chapter 2, we study a Lifting Lemma that can prove shuffle compatibility for some statistics on circular permutations based on known results for those statistics on linear permutations. One well-studied permutation statistic is the {\em peak set}, which is the set of all indices of a permutation where an element is adjacent to two smaller elements. Primarily spearheaded by Davis, Nelson, Petersen, and Tenner, there has been recent interest in studying an analogue of this statistic known as the {\em pinnacle set}, which are the values of the elements at the indices of the peak set. Davis et al.\ proposed a number of unanswered questions about this statistic, which later led to a series of papers on the topic. In Chapter 3 we will present multiple results that attempt to answer some of these questions, including some original formulas and also some alternate combinatorial proofs of known results. These will include a bijection for counting the number of sets that could be the pinnacle set of some permutation, formulas and recursions for counting the number of permutations with a given pinnacle set, along with a new proof for a weighted sum of those numbers, and a recursion for counting the number of distinct orderings in which the elements of a pinnacle set can appear within a permutation.
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