### Abstract

In this extended abstract we consider the poset of weighted partitions ∏^{w}_{n} , introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of ∏^{w}_{n} provide a generalization of the lattice ∏_{n} of partitions, which we show possesses many of the well-known properties of ∏_{n}. In particular, we prove these intervals are EL-shellable, we compute the M̈obius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted O_{n}-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of ∏^{w}_{n}has a nice factorization analogous to that of ∏_{n}.

Original language | English (US) |
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Pages (from-to) | 1029-1040 |

Number of pages | 12 |

Journal | Discrete Mathematics and Theoretical Computer Science |

State | Published - Nov 18 2013 |

Event | 25th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2013 - Paris, France Duration: Jun 24 2013 → Jun 28 2013 |

### Keywords

- Free lie algebra
- Partitions
- Poset topology
- Rooted trees

### ASJC Scopus subject areas

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

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## Cite this

*Discrete Mathematics and Theoretical Computer Science*, 1029-1040.