### Abstract

We consider the poset of weighted partitions Π^{w} _{n}, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π^{w} _{n} provide a generalization of the lattice Π_{n} of partitions, which we show possesses many of the well-known properties of Π_{n}. In particular, we prove these intervals are EL-shellable, we show that the bius invariant of each maximal interval is given up to sign by the number of rooted trees on node set {1, 2,…, n} having a fixed number of descents, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted S_{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 Π^{w} _{n} has a nice factorization analogous to that of Π_{n}.

Original language | English (US) |
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Pages (from-to) | 6779-6818 |

Number of pages | 40 |

Journal | Transactions of the American Mathematical Society |

Volume | 368 |

Issue number | 10 |

DOIs | |

State | Published - Oct 1 2016 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*368*(10), 6779-6818. https://doi.org/10.1090/tran/6483