## Abstract

The depth statistic was defined for every Coxeter group in terms of factorizations of its elements into product of reflections. Essentially, the depth gives the minimal path cost in the Bruaht graph, where the edges have prescribed weights. We present an algorithm for calculating the depth of a signed permutation which yields a simple formula for this statistic. We use our algorithm to characterize signed permutations having depth equal to length. These are the fully commutative top-and-bottom elements defined by Stembridge. We finally give a characterization of the signed permutations in which the reflection length coincides with both the depth and the length.

Original language | English |
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Pages (from-to) | 913-924 |

Number of pages | 12 |

Journal | Discrete Mathematics and Theoretical Computer Science |

State | Published - 2015 |

Externally published | Yes |

Event | 27th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2015 - Daejeon, Korea, Republic of Duration: 6 Jul 2015 → 10 Jul 2015 |

### Bibliographical note

Publisher Copyright:© 2015 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.

## Keywords

- Bruhat graph
- Coxeter groups
- Depths
- Length
- Reflections