Errate slip inserted / Bibliography: leaves 130-134 / vi, 134 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Statistics, 1988

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Prasad Tetali; Committee Member: Dana Randall; Committee Member: Robin Thomas; Committee Member: Vijay Vazirani; Committee Member: William T. Trotter

Thesis (Ph.D.)--York University, 2007. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 109-110). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR32041

The five Mathieu permutation groups M₁₁, M₁₂M₂₂,M₂₃ and M₂₄ are constructed and the involutions (elements of order two) of these groups are classified according to the number of letters they fix. It is shown that in M₁₂ ah involution fixes no letters or four letters, while in M₂₄ an involution fixes zero or eight letters. It is also shown that in each of the Mathieu groups, all the irregular involutions are conjugate and that in M₁₂ all the regular involutions are conjugate. The orders of the centralizers of the involutions are calculated and it is shown that no regular involution lies in the centre of a 2-Sylow subgroup. Most of the results are obtained by calculating directly the form a permutation must take in order to have a certain property and then finding one or all the permutations of this form. / Science, Faculty of / Mathematics, Department of / Graduate

The study of pattern classes is the study of the involvement order on finite permutations. This order can be traced back to the work of Knuth. In recent years the area has attracted the attention of many combinatoralists and there have been many structural and enumerative developments. We consider permutations classes defined in three different ways and demonstrate that asking the same fixed questions in each case motivates a different view of involvement. Token passing networks encourage us to consider permutations as sequences of integers; grid classes encourage us to consider them as point sets; picture classes, which are developed for the first time in this thesis, encourage a purely geometrical approach. As we journey through each area we present several new results. We begin by studying the basic definitions of a permutation. This is followed by a discussion of the questions one would wish to ask of permutation classes. We concentrate on four particular areas: partial well order, finite basis, atomicity and enumeration. Our third chapter asks these questions of token passing networks; we also develop the concept of completeness and show that it is decidable whether or not a particular network is complete. Next we move onto grid classes, our analysis using generic sets yields an algorithm for determining when a grid class is atomic; we also present a new and elegant proof which demonstrates that certain grid classes are partially well ordered. The final chapter comprises the development and analysis of picture classes. We completely classify and enumerate those permutations which can be drawn from a circle, those which can be drawn from an X and those which can be drawn from some convex polygon. We exhibit the first uncountable set of closed classes to be found in a natural setting; each class is drawn from three parallel lines. We present a permutation version of the famous `happy ending' problem of Erdös and Szekeres. We conclude with a discussion of permutation classes in higher dimensional space.

In this thesis we have presented original homomorphic images of permutations and monomial progenitors. In some cases we have used the double coset enumeration tech- nique to construct the images and for all of the homomorphic images that we have discovered, the isomorphism type of each group is given. The homomorphic images discovered include Linear groups, Alternating groups, and two sporadic simple groups J1 and J2X2 where J1 is the smallest Janko group and J2 is the second Janko sporadic group.

We consider the problem of counting <i>transitive factorizations</i> of permutations; that is, we study tuples (&sigma;<i>r</i>,. . . ,&sigma;1) of permutations on {1,. . . ,<i>n</i>} such that (1) the product &sigma;<i>r</i>. . . &sigma;1 is equal to a given target permutation &pi;, and (2) the group generated by the factors &sigma;<i>i</i> acts transitively on {1,. . . ,<i>n</i>}. This problem is widely known as the <i>Hurwitz Enumeration Problem</i>, since an encoding due to Hurwitz shows it to be equivalent to the enumeration of connected branched coverings of the sphere by a surface of given genus with specified branching. Much of our work concerns the enumeration of transitive factorizations of permutations into a minimal number of transposition factors. This problem has received considerable attention, and a formula for the number <i>c</i>(&pi;) of such factorizations of an arbitrary permutation &pi; has been derived through various means. The formula is remarkably simple, being a product of well-known combinatorial numbers, but no bijective proof of it is known except in the special case where &pi; is a full cycle. A major goal of this thesis is to provide further combinatorial rationale for this formula. We begin by introducing an encoding of factorizations (into transpositions) as edge-labelled maps. Our central result is a bijection that allows trees to be "pruned" from such maps. This is shown to explain the appearance of factors of the form <i>k^k</i> in the aforementioned formula for <i>c</i>(&pi;). It also has the effect of shifting focus to the combinatorics of smooth maps (<i>i. e. </i> maps without vertices of degree one). By providing decompositions for certain smooth planar maps, we are able to give combinatorial evaluations of <i>c</i>(&pi;) when &pi; is composed of up to three cycles. Many of these results are generalized to factorizations in which the factors are cycles of any length. We also investigate the <i>Double Hurwitz Problem</i>, which calls for the enumeration of factorizations whose leftmost factor is of specified cycle type, and whose remaining factors are transpositions. Finally, we extend our methods to the enumeration of factorizations up to an equivalence relation induced by possible commutations between adjacent factors.

We consider the problem of counting <i>transitive factorizations</i> of permutations; that is, we study tuples (&sigma;<i>r</i>,. . . ,&sigma;1) of permutations on {1,. . . ,<i>n</i>} such that (1) the product &sigma;<i>r</i>. . . &sigma;1 is equal to a given target permutation &pi;, and (2) the group generated by the factors &sigma;<i>i</i> acts transitively on {1,. . . ,<i>n</i>}. This problem is widely known as the <i>Hurwitz Enumeration Problem</i>, since an encoding due to Hurwitz shows it to be equivalent to the enumeration of connected branched coverings of the sphere by a surface of given genus with specified branching. Much of our work concerns the enumeration of transitive factorizations of permutations into a minimal number of transposition factors. This problem has received considerable attention, and a formula for the number <i>c</i>(&pi;) of such factorizations of an arbitrary permutation &pi; has been derived through various means. The formula is remarkably simple, being a product of well-known combinatorial numbers, but no bijective proof of it is known except in the special case where &pi; is a full cycle. A major goal of this thesis is to provide further combinatorial rationale for this formula. We begin by introducing an encoding of factorizations (into transpositions) as edge-labelled maps. Our central result is a bijection that allows trees to be "pruned" from such maps. This is shown to explain the appearance of factors of the form <i>k^k</i> in the aforementioned formula for <i>c</i>(&pi;). It also has the effect of shifting focus to the combinatorics of smooth maps (<i>i. e. </i> maps without vertices of degree one). By providing decompositions for certain smooth planar maps, we are able to give combinatorial evaluations of <i>c</i>(&pi;) when &pi; is composed of up to three cycles. Many of these results are generalized to factorizations in which the factors are cycles of any length. We also investigate the <i>Double Hurwitz Problem</i>, which calls for the enumeration of factorizations whose leftmost factor is of specified cycle type, and whose remaining factors are transpositions. Finally, we extend our methods to the enumeration of factorizations up to an equivalence relation induced by possible commutations between adjacent factors.

Thesis (Ph. D.)--Missouri University of Science and Technology, 2008. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed May 9, 2008) Includes bibliographical references (p. 71-73).

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 115-119).