Jump numbers, hyperrectangles and Carlitz compositions

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 1998.A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg 1998Let A = (aij) be an m x n matrix. There is a natural way to associate a poset PA with A. A jump in a linear extension of PA is a pair of consecutive elements which are incomparable in Pa. The jump number of A is the minimum number of jumps in any linear extension of PA. The maximum jump number over a class of n x n matrices of zeros and ones with constant row and column sum k, M (n, k), has been investigated in Chapter 2 and 3. Chapter 2 deals with extremization problems concerning M (n ,k). In Chapter 3, we obtain the exact values for M (11,k). M(n,Q), M (n,n-3) and M(n,n-4). The concept of frequency hyperrectangle generalizes the concept of latin square. In Chapter 4 we derive a bound for the maximum number of mutually orthogonal frequency hyperrectangles. Chapter 5 gives two algorithms to construct mutually orthogonal frequency hyperrectangles. Chapter 6 is devoted to some enumerative results about Carlitz compositions (compositions with different adjacent parts)