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.

This dissertation is divided into four chapters. In Chapter 2 the equivariant homology groups of upper order ideals in the partition lattice are computed. The homology groups of these filters are written in terms of border strip Specht modules as well as in terms of links in an associated complex in the lattice of compositions. The classification is used to reproduce topological calculations of many well-studied subcomplexes of the partition lattice, including the d-divisible partition lattice and the Frobenius complex. In Chapter 3 the box polynomial B_{m,n}(x) is defined in terms of all integer partitions that fit in an m by n box. The real roots of the box polynomial are completely characterized, and an asymptotically tight bound on the norms of the complex roots is also given. An equivalent definition of the box polynomial is given via applications of the finite difference operator Delta to the monomial x^{m+n}. The box polynomials are also used to find identities counting set partitions with all even or odd blocks, respectively. Chapter 4 extends results from Chapter 3 to give combinatorial proofs for the ordinary generating function for set partitions with all even or all odd block sizes, respectively. This is achieved by looking at a multivariable generating function analog of the Stirling numbers of the second kind using restricted growth words. Chapter 5 introduces a colored variant of the ordered partition lattice, denoted Q_n^{\alpha}, as well an associated complex known as the alpha-colored permutahedron, whose face poset is Q_n^\alpha. Connections between the Eulerian polynomials and Stirling numbers of the second kind are developed via the fibers of a map from Q_n^{\alpha} to the symmetric group on n-elements

Runtime enforcement mechanisms are an important and well-employed method for ensuring an execution only exhibits acceptable behavior, as dictated by a security policy. Wherever interaction occurs between two or more parties that do not completely trust each other, it is most often the case that a runtime enforcement mechanism is between them in some form, monitoring the exchange. Considering the ubiquity of such scenarios in the computing world, there has been an increased effort to build formal models of runtime monitors that closely capture their capabilities so that their effectiveness can be analysed more precisely. While models have grown more faithful to their real-life counterparts, is- sues concerning complexity and manageability (a common concern for software engineers) of centralized policies remains to be fully addressed. The goal of this thesis is to provide a principled approach to policy construction that is modular, intuitive, and backed by formal methods. This thesis introduces a class of policy combinators adequate for use with runtime en- forcement policies and analyses a particular instance of them called Static Committee Com- binators (SCCs). SCCs present a model of policy composition where combinators act as committees that vote on events passing through the monitor. They were conceptualized in collaboration with Jay Ligatti and Daniel Lomsak. The general class of combinators are called Static Decision Combinators (SDCs), which share key features with SCCs such as allowing combinators to respond with alternative events when polled, in addition to re- sponding with grants or denials. SDCs treat the base-level policies they compose as black boxes, which helps decouple the system of combinators from the underlying policy model. The base policies could be modelled by automata but the combinators would not maintain their own state, being "static". This allows them to be easily defined and understood using truth tables, as well as analysed using logic tools. In addition to an analysis of SDCs and SCCs, we provide useful examples and a reusable combinator library.

Induced graphs are used to describe the structure of a graph, one such type of induced graph that has been studied are long paths. In this thesis we show a way to represent such graphs in terms of an array with two colors and a labeled graph. Using this representation and the techniques of Polya counting we will then be able to get upper and lower bounds for graphs containing a long path as an induced subgraph. In particular, if we let P(n,k) be the number of graphs on n+k vertices which contains P_n, a path on n vertices, as an induced subgraph then using our upper and lower bounds for P(n,k) we will show that for any fixed value of k that P(n,k)~2^(nk+k_C_2)/(2k!).

Yes / It is shown that the probability generating function of a lifetime random variable T on a finite lattice with polynomial failure rate satisfies a certain differential equation. The interrelationship with Markov chain theory is highlighted. The differential equation gives rise to a system of differential equations which, when inverted, can be used in the limit to express the polynomial coefficients in terms of the factorial moments of T. This then can be used to estimate the polynomial coefficients. Some special cases are worked through symbolically using Computer Algebra. A simulation study is used to validate the approach and to explore its potential in the reliability context.

Made available in DSpace on 2017-07-21T20:56:33Z (GMT). No. of bitstreams: 1 Pedro Kiochi Kondo.pdf: 1227541 bytes, checksum: daffb8a8bc299356bce288603753944c (MD5) Previous issue date: 2014-09-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work some topics of the Discrete or Finite Calculus are developed. In particular, we study difference operators, factorial powers, Stirling numbers of the first and second type, the Newton’s formula of differences, the fundamental theorem of the Finite Calculus, the summation process, and the Bernoulli numbers and Bernoulli polynomials. Then we show the effectiveness of the theory for the calculation of closed formulas for the value of many finite sums. We also study the classical problem of obtaining the polynomials which express the value of the sums of powers of natural numbers. / Neste trabalho desenvolvemos alguns tópicos do Cálculo Discreto ou Finito. Em particular, estudamos operadores de diferenças, potências fatoriais, números de Stirling do primeiro e do segundo tipo, a fórmula de diferenças de Newton, o teorema fundamental do Cálculo Finito, o processo de somação e os números e polinômios de Bernoulli. Mostramos então a eficácia da teoria no cálculo de fórmulas fechadas para o valor de diversas somas finitas. Também estudamos o problema clássico de obter os polinômios que expressam o valor de somas de potências de números naturais.

Cette thèse se divise en 2 grandes parties indépendantes ; la première traitant des nombres de Jacobi-Stirling, la seconde abordant les nombres d’Entringer. La première partie introduit les nombres de Jacobi-Stirling de seconde et de première espèce comme coefficients algébriques dans des relations polynomiales. Nous donnons des interprétations combinatoires de ces nombres, en termes de partitions d’ensembles et de quasi-permutations pour les nombres de seconde espèce, et en termes de permutations pour les nombres de première espèce. Nous étudions également les fonctions génératrices diagonales de ces familles de nombres, ainsi qu’une de leur généralisation sur le modèle des r-nombres de Stirling. La seconde partie introduit les nombres d’Entringer à l’aide de leur interprétation en termes de permutations alternantes. Nous étudions les différentes formules de récurrence vérifiées par ces nombres et généralisons ces résultats à l’aide d’un q-analogue utilisant la statistique d’inversion. Nous verrons également que ces résultats peuvent être étendus à des permutations de forme donnée quelconque. Enfin, nous définissons la notion de famille d’Entringer, et établissons des bijections entre certaines de ces familles. En particulier, nous établissons une bijection reliant les permutations alternantes de premier terme fixé, aux arbres binaires croissants dont l’extrémité du chemin minimal est fixée. / This thesis is constructed in two main independant parts ; the first one dealing with the numbers of Jacobi-Stirling, the second one tackling the numbers of Entringer. The first part introduces the numbers of Jacobi-Stirling of the second kind and of the first kind, as algebraic coefficients in some polynomial relations. We give some combinatorial interpretations of these numbers, in terms of set partitions and quasi-permutations for the numbers of the second kind, and in terms of permutations for the numbers of the first kind. We also study the diagonal generating functions of these sequences of numbers, and one of their generalization based on the model of r-Stirling numbers. The second part introduces the numbers of Entringer with their interpretation in terms of alternating permutations. We study the different recurrences formulas satisfied by these numbers, and refine these results with a q-analogue using the inversion statistic. We also note that these results can be extend to permutations with any fixed shape. Finally, we define the notion of Entringer family, and provide bijections between some of these families. In particular, we establish a bijection between the alternating permutations with fixed given value, and the binary increasing trees such that the end-point of the minimal path is fixed.

In its theoretical part, this thesis sums up the basic knowledge concerning permutations. Besides the representation of permutations and determination of their fundamental characteristics, the theoretical part is, first of all, aimed at results concerning the decomposition of permutations into disjoint cycles and at finding the number of permutations with a certain characteristic. We introduce the fundamental bijection that is useful for solving many problems concerning the permutations. Further on, we focus on the number of permutations without a fixed point, Eulerian numbers expressing the number of permutations with a given number of descents, and the number of permutations with a given number of excedances, Stirling numbers of the first kind expressing the number of permutations with a given number of cycles, and Catalan numbers representing the number of permutations avoiding a chosen pattern of length three. Attention is also paid to the Gilbreath permutations and their characteristics. The practical part consists of 14 solved problems. The solutions rely on the results presented in the theoretical part, and there are deduced some further interesting results concerning random permutations.