In this dissertation we discuss three problems. We first show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. We also give a bijective argument showing the (q, t)-Stirling numbers of the first and second kind are orthogonal. In the second part we give combinatorial proofs of q-Stirling identities via restricted growth words. This includes new proofs of the generating function of q-Stirling numbers of the second kind, the q-Vandermonde convolution for Stirling numbers and the q-Frobenius identity. A poset theoretic proof of Carlitz’s identity is also included. In the last part we discuss a new expression for q-binomial coefficients based on the weighting of certain 01-permutations via a new bistatistic related to the major index. We also show that the bistatistics between the inversion number and major index are equidistributed. We generalize this idea to q-multinomial coefficients evaluated at negative q values. An instance of the cyclic sieving phenomenon related to flags of unitary spaces is also studied.

In this dissertation we will look at properties of two different posets from different perspectives. The first poset is the Rees product of the face lattice of the n-cube with the chain. Specifically we study the Möbius function of this poset. Our proof techniques include straightforward enumeration and a bijection between a set of labeled augmented skew diagrams and barred signed permutations which label the maximal chains of this poset. Because the Rees product of this poset is Cohen-Macaulay, we find a basis for the top homology group and a representation of the top homology group over the symmetric group both indexed by the set of labeled augmented skew diagrams. We also show that the Möbius function of the Rees product of a graded poset with the t-ary tree and the Rees product of its dual with the t-ary tree coincide. We discuss labelings for Rees and Segre products in general, particularly the Rees product of the face lattice of a polytope with the chain. We also look at cases where the Möbius function of a poset is equal to the permanent of a matrix and we consider local h-vectors for the barycentric subdivision of the n-cube. In each section we state open conjectures. The second poset in this dissertation is the Dowling lattice. In particular we look at the k = 1 case, that is, the partition lattice. We study inequalities on the flag vector of the partition lattice via a weighted boustrophedon transform and determine a more generalized version for the Dowling lattice. We generalize a determinantal formula of Niven and conclude with conjectures and avenues of study.

The game of Subset Take-Away is an impartial combinatorial game posed by David Gale in 1974. The game can be played on various discrete structures, including but not limited to graphs, hypergraphs, polygonal complexes, and partially ordered sets. While a universal winning strategy has yet to be found, results have been found in certain cases. In 2003 R. Riehemann focused on Subset Take-Away on bipartite graphs and produced a complete game analysis by studying nim-values. In this work, we extend the notion of Take-Away on a bipartite graph to Take-Away on particular hypergraphs, namely oddly-uniform hypergraphs and evenly-uniform hypergraphs whose vertices satisfy a particular coloring condition. On both structures we provide a complete game analysis via nim-values. From here, we consider different discrete structures and slight variations of the rules for Take-Away to produce some interesting results. Under certain conditions, polygonal complexes exhibit a sequence of nim-values which are unbounded but have a well-behaved pattern. Under other conditions, the nim-value of a polygonal complex is bounded and predictable based on information about the complex itself. We introduce a Take-Away variant which we call “Take-As-Much-As-You-Want”, and we show that, again, nim-values can grow without bound, but fortunately they are very easily described for a given graph based on the total number of vertices and edges of the graph. Finally we consider Take-Away on a specific type of partially ordered set which we call a rank-complete poset. We have results, again via an analysis of nim-values, for Take-Away on a rank-complete poset for both ordinary play as well as misère play.

This thesis studies data structures in the word-RAM and bitprobe models, with an emphasis on space efficiency. In the word-RAM model of computation the space cost of a data structure is measured in terms of the number of w-bit words stored in memory, and the cost of answering a query is measured in terms of the number of read, write, and arithmetic operations that must be performed. In the bitprobe model, like the word-RAM model, the space cost is measured in terms of the number of bits stored in memory, but the query cost is measured solely in terms of the number of bit accesses, or probes, that are performed. First, we examine the problem of succinctly representing a partially ordered set, or poset, in the word-RAM model with word size Theta(lg n) bits. A succinct representation of a combinatorial object is one that occupies space matching the information theoretic lower bound to within lower order terms. We show how to represent a poset on n vertices using a data structure that occupies n^2/4 + o(n^2) bits, and can answer precedence (i.e., less-than) queries in constant time. Since the transitive closure of a directed acyclic graph is a poset, this implies that we can support reachability queries on an arbitrary directed graph in the same space bound. As far as we are aware, this is the first representation of an arbitrary directed graph that supports reachability queries in constant time, and stores less than n choose 2 bits. We also consider several additional query operations. Second, we examine the problem of supporting range queries on strings of n characters (or, equivalently, arrays of n elements) in the word-RAM model with word size Theta(lg n) bits. We focus on the specific problem of answering range majority queries: i.e., given a range, report the character that is the majority among those in the range, if one exists. We show that these queries can be supported in constant time using a linear space (in words) data structure. We generalize this result in several directions, considering various frequency thresholds, geometric variants of the problem, and dynamism. These results are in stark contrast to recent work on the similar range mode problem, in which the query operation asks for the mode (i.e., most frequent) character in a given range. The current best data structures for the range mode problem take soft-Oh(n^(1/2)) time per query for linear space data structures. Third, we examine the deterministic membership (or dictionary) problem in the bitprobe model. This problem asks us to store a set of n elements drawn from a universe [1,u] such that membership queries can be always answered in t bit probes. We present several new fully explicit results for this problem, in particular for the case when n = 2, answering an open problem posed by Radhakrishnan, Shah, and Shannigrahi [ESA 2010]. We also present a general strategy for the membership problem that can be used to solve many related fundamental problems, such as rank, counting, and emptiness queries. Finally, we conclude with a list of open problems and avenues for future work.

In 2001, Fishburn, Tanenbaum, and Trenk published a pair of papers that introduced the notions of linear and weak discrepancy of a partially ordered set or poset. Linear discrepancy for a poset is the least k such that for any ordering of the points in the poset there is a pair of incomparable points at least distance k away in the ordering. Weak discrepancy is similar to linear discrepancy except that the distance is observed over weak labelings (i.e. two points can have the same label if they are incomparable, but order is still preserved). My thesis gives a variety of results pertaining to these properties and other forms of discrepancy in posets. The first chapter of my thesis partially answers a question of Fishburn, Tanenbaum, and Trenk that was to characterize those posets with linear discrepancy two. It makes the characterization for those posets with width two and references the paper where the full characterization is given. The second chapter introduces the notion of t-discrepancy which is similar to weak discrepancy except only the weak labelings with at most t copies of any label are considered. This chapter shows that determining a poset's t-discrepancy is NP-Complete. It also gives the t-discrepancy for the disjoint sum of chains and provides a polynomial time algorithm for determining t-discrepancy of semiorders. The third chapter presents another notion of discrepancy namely total discrepancy which minimizes the average distance between incomparable elements. This chapter proves that finding this value can be done in polynomial time unlike linear discrepancy and t-discrepancy. The final chapter answers another question of Fishburn, Tanenbaum, and Trenk that asked to characterize those posets that have equal linear and weak discrepancies. Though determining the answer of whether the weak discrepancy and linear discrepancy of a poset are equal is an NP-Complete problem, the set of minimal posets that have this property are given. At the end of the thesis I discuss two other open problems not mentioned in the previous chapters that relate to linear discrepancy. The first asks if there is a link between a poset's dimension and its linear discrepancy. The second refers to approximating linear discrepancy and possible ways to do it.

This thesis studies data structures in the word-RAM and bitprobe models, with an emphasis on space efficiency. In the word-RAM model of computation the space cost of a data structure is measured in terms of the number of w-bit words stored in memory, and the cost of answering a query is measured in terms of the number of read, write, and arithmetic operations that must be performed. In the bitprobe model, like the word-RAM model, the space cost is measured in terms of the number of bits stored in memory, but the query cost is measured solely in terms of the number of bit accesses, or probes, that are performed. First, we examine the problem of succinctly representing a partially ordered set, or poset, in the word-RAM model with word size Theta(lg n) bits. A succinct representation of a combinatorial object is one that occupies space matching the information theoretic lower bound to within lower order terms. We show how to represent a poset on n vertices using a data structure that occupies n^2/4 + o(n^2) bits, and can answer precedence (i.e., less-than) queries in constant time. Since the transitive closure of a directed acyclic graph is a poset, this implies that we can support reachability queries on an arbitrary directed graph in the same space bound. As far as we are aware, this is the first representation of an arbitrary directed graph that supports reachability queries in constant time, and stores less than n choose 2 bits. We also consider several additional query operations. Second, we examine the problem of supporting range queries on strings of n characters (or, equivalently, arrays of n elements) in the word-RAM model with word size Theta(lg n) bits. We focus on the specific problem of answering range majority queries: i.e., given a range, report the character that is the majority among those in the range, if one exists. We show that these queries can be supported in constant time using a linear space (in words) data structure. We generalize this result in several directions, considering various frequency thresholds, geometric variants of the problem, and dynamism. These results are in stark contrast to recent work on the similar range mode problem, in which the query operation asks for the mode (i.e., most frequent) character in a given range. The current best data structures for the range mode problem take soft-Oh(n^(1/2)) time per query for linear space data structures. Third, we examine the deterministic membership (or dictionary) problem in the bitprobe model. This problem asks us to store a set of n elements drawn from a universe [1,u] such that membership queries can be always answered in t bit probes. We present several new fully explicit results for this problem, in particular for the case when n = 2, answering an open problem posed by Radhakrishnan, Shah, and Shannigrahi [ESA 2010]. We also present a general strategy for the membership problem that can be used to solve many related fundamental problems, such as rank, counting, and emptiness queries. Finally, we conclude with a list of open problems and avenues for future work.