We provide a particular measure for the degree to which an arbitrary composition deviates from increasing sorted order. The application of such a measure to the transport industry is given in the introduction. In order to obtain this measure, we define a statistic called the number of pushes in an arbitrary composition (which is required to produce sorted order) and obtain a generating function for this. The concept of a push is a geometrical one and leads naturally to several dependant concepts which are investigated. These are the number of cells which do not move in the pushing process and the number of cells that coincide before and after the pushing process (a number not less than those that do not move). The concept of a push leads to combining certain single pushes in a natural way which we define as a frictionless push. A generating function for these is also developed. The underlying geometry of the process also leads naturally to counting the largest first component of arbitrary compositions that are already in a sorted order. We provide a generating function for this.
Crystallography of Coaxial and Scroll Nanotubes of Arbitrary Composition
