13 research outputs found
Derangements in a Ferrers Board
The classic derangement question of counting the number of derangements for n objects from some initial permutation of the objects was first considered by de Montfort in 1708. A particular recasting of a permutation allows us to place any permutation onto an n x n board, from which certain properties of derangements may be understood. This research extends the classic derangement question to the more general Ferrers board, which is an n x n board with a missing section in the lower-right corner. Various properties of the derangement numbers for these more general boards are stated and proven in the course of this work
Investigation of a Markov Chain on Ferrers Boards
This thesis is an investigation of some of the basic combinatorial, algebraic and probabilistic properties of a Markov chain on Ferrers Boards (i.e., a Markov chain whose states are permutations on a given Ferrers Board). This is an extension of extensive work done over the last fifty years to understand the properties of a Markov chain known as the Tsetlin library. We will review the extensive literature surrounding the Tsetlin library, which also allows for the problem to be contextualized as a particularly nice model of a procedure for searching a database of files. Some of the specific questions we will explore include the transitivity of the Tsetlin library (in fact, we will prove that the extended library is transitive and at most n steps are needed to reach any state from an arbitrarily chosen state); the Tsetlin library’s relation to permutation inversions and some other combinatorial statistics; and finally the computation of the Tsetlin library’s stationary distribution and eigenvalues in some easy cases.
Although our analysis of the combinatorial aspects of the extended Tsetlin library is complete, we have been unable to fully describe the probabilistic aspects of the Tsetlin library. We are able to describe the stationary distribution for specific easy cases, but further analysis for more complicated cases has proven difficult. Computations have been done using the mathematical software Maple to determine if any patterns may be discerned from specific examples of the more complicated cases. However, the data indicates that the actual stationary distribution differs from our conjectured formula for the stationary distribution, which gives a need for further analysis in future work. We have also not been able to describe the eigenvalues or convergence to stationary for even the simplest Ferrers boards, but we do have various computations which we hope will be the basis for future exploration of these topics
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The life and work of Major Percy Alexander MacMahon
This thesis describes the life and work of the mathematician Major Percy Alexander MacMahon (1854 - 1929). His early life as a soldier in the Royal Artillery and events which led to him embarking on a career in mathematical research and teaching are dealt with in the first two chapters. Succeeding chapters explain the work in invariant theory and partition theory which brought him to the attention of the British mathematical community and eventually resulted in a Fellowship of the Royal Society, the presidency of the London Mathematical Society, and the award of three prestigious mathematical medals and four honorary doctorates. The development and importance of his recreational mathematical work is traced and discussed. MacMahon's career in the Civil Service as Deputy Warden of the Standards at the Board of Trade is also described. Throughout the thesis, his involvement with the British Association for the Advancement of Science and other scientific organisations is highlighted. The thesis also examines possible reasons why MacMahon's work, held in very high regard at the time, did not lead to the lasting fame accorded to some of his contemporaries. Details of his personal and social life are included to give a picture of MacMahon as a real person working hard to succeed in a difficult context