## Abstract

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.

Original language | English |
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Article number | 2050079 |

Journal | Discrete Mathematics, Algorithms and Applications |

Volume | 12 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2020 |

### Bibliographical note

Funding Information:Charlotte Brennan and Arnold Knopfmacher mentioned that this material is based upon work supported by the National Research Foundation Under Grant Numbers 86329 and 81021, respectively.

Publisher Copyright:

© 2020 World Scientific Publishing Company.

## Keywords

- Compositions
- asymptotics
- generating function

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics