## Abstract

In this paper, we consider a new statistic on the set S_{n} of permutations of length n which records the number of vertical edges within their bargraph representations. In-deed, we determine the distribution of this statistic on a class of multi-permutations having S_{n} as a subset. We compute an explicit formula for the total number of vertical edges within all members of this class and also for the number of interior vertices in the corresponding bargraphs. A generating function formula is determined and some closely related statistics are considered. Finally, we show that the total number of vertical edges in all the members of S_{n} equals the number of runs in the members of S_{n+1} by constructing an explicit bijection which applies more generally to multi-permutations.

Original language | English |
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Pages (from-to) | 279-290 |

Number of pages | 12 |

Journal | Journal of Automata, Languages and Combinatorics |

Volume | 25 |

Issue number | 4 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Publisher Copyright:© Institut für Informatik · Justus-Liebig-Universität Giessen.

## Keywords

- Bargraphs
- Generating functions
- Multi-set permutations
- Permutation statistics

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics