Jump numbers, hyperrectangles and Carlitz compositions

Abstract

Let 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).