### Abstract

We prove that the external activity complex Act<(M) of a matroid is shellable. In fact, we show that every linear extension of Las Vergnas's external/internal order <_{ext/int} on M provides a shelling of Act<(M). We also show that every linear extension of Las Vergnas's internal order <int on M provides a shelling of the independence complex IN(M). As a corollary, Act<(M) and M have the same h-vector. We prove that, after removing its cone points, the external activity complex is contractible if M contains U3,1 as a minor, and a sphere otherwise.

Original language | English (US) |
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Pages (from-to) | 73-82 |

Number of pages | 10 |

Journal | Discrete Mathematics and Theoretical Computer Science |

State | Published - 2016 |

Externally published | Yes |

Event | 28th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2016 - Vancouver, Canada Duration: Jul 4 2016 → Jul 8 2016 |

### Keywords

- External/internal order
- Matroids
- Shellability
- Simplicial complexes

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)
- Discrete Mathematics and Combinatorics

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## Cite this

Ardila, F., Castillo, F., & Samper, J. A. (2016). The topology of the external activity complex of a matroid.

*Discrete Mathematics and Theoretical Computer Science*, 73-82.