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<p>This thesis is about a generalization of permutation theory. The concept of pattern avoidance in permutation matrices is investigated in a larger class of matrices - the alternating sign matrices. The main result is that the set of alternating sign matrices avoiding the pattern 132, is counted by the large Schröder numbers. An algebraic and a bijective proof is presented. Another class is shown to be counted by every second Fibonacci number. Further research in this new area of combinatorics is discussed.</p>

This thesis is about a generalization of permutation theory. The concept of pattern avoidance in permutation matrices is investigated in a larger class of matrices - the alternating sign matrices. The main result is that the set of alternating sign matrices avoiding the pattern 132, is counted by the large Schröder numbers. An algebraic and a bijective proof is presented. Another class is shown to be counted by every second Fibonacci number. Further research in this new area of combinatorics is discussed.

International Telemetering Conference Proceedings / October 20-23, 2003 / Riviera Hotel and Convention Center, Las Vegas, Nevada / An artifact of QPSK modems is ambiguity of the recovered data. There are four variations of the output data for a given input data stream. All are equally probable. To resolve this ambiguity, the QPSK data streams can be differentially encoded before modulation and differentially decoded after demodulation. The encoder maps each input data pair to a phase angle change of the QPSK carrier. In the demodulator, the inverse is performed - each phase change of the input QPSK carrier is mapped to an output data pair. This paper discusses a very simple and unique differential encoder/decoder that handles all possible data pair/phase change permutations.

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Conditioning on the observed data is an important and flexible design principle for statistical test procedures. Although generally applicable, permutation tests currently in use are limited to the treatment of special cases, such as contingency tables or K-sample problems. A new theoretical framework for permutation tests opens up the way to a unified and generalized view. We argue that the transfer of such a theory to practical data analysis has important implications in many applications and requires tools that enable the data analyst to compute on the theoretical concepts as closely as possible. We re-analyze four data sets by adapting the general conceptual framework to these non-standard inference procedures and utilizing the coin add-on package in the R system for statistical computing to show what one can gain from going beyond the `classical' test procedures. / Series: Research Report Series / Department of Statistics and Mathematics

Seriation, i.e., finding a suitable linear order for a set of objects given data and a loss or merit function, is a basic problem in data analysis. Caused by the problem's combinatorial nature, it is hard to solve for all but very small sets. Nevertheless, both exact solution methods and heuristics are available. In this paper we present the package seriation which provides an infrastructure for seriation with R. The infrastructure comprises data structures to represent linear orders as permutation vectors, a wide array of seriation methods using a consistent interface, a method to calculate the value of various loss and merit functions, and several visualization techniques which build on seriation. To illustrate how easily the package can be applied for a variety of applications, a comprehensive collection of examples is presented.

We will examine progenitors. We begin with progenitors of the from $m^{*n} : N$ where $m^{*n}$ is a free group and $N$ is a permutation group of degree $n$. Now $m^{*n} : N$ is a group of infinite order so we will factor by the necessary relations to get the finite homomorphic images. We construct these groups through the method of double coset enumeration paying special attention to the proving of each relation.

We introduce and explore an approach to estimating statistical significance of classification accuracy, which is particularly useful in scientific applications of machine learning where high dimensionality of the data and the small number of training examples render most standard convergence bounds too loose to yield a meaningful guarantee of the generalization ability of the classifier. Instead, we estimate statistical significance of the observed classification accuracy, or the likelihood of observing such accuracy by chance due to spurious correlations of the high-dimensional data patterns with the class labels in the given training set. We adopt permutation testing, a non-parametric technique previously developed in classical statistics for hypothesis testing in the generative setting (i.e., comparing two probability distributions). We demonstrate the method on real examples from neuroimaging studies and DNA microarray analysis and suggest a theoretical analysis of the procedure that relates the asymptotic behavior of the test to the existing convergence bounds.