### 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) |
---|---|

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

**On the (CO)homology of the poset of weighted partitions.** / González D’león, Rafael S.; Galloway, Michelle L.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 368, no. 10, pp. 6779-6818. https://doi.org/10.1090/tran/6483

}

TY - JOUR

T1 - On the (CO)homology of the poset of weighted partitions

AU - González D’león, Rafael S.

AU - Galloway, Michelle L

PY - 2016/10/1

Y1 - 2016/10/1

N2 - We consider the poset of weighted partitions Πw n, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Πw n provide a generalization of the lattice Πn of partitions, which we show possesses many of the well-known properties of Πn. 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 Sn-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 Πw n has a nice factorization analogous to that of Πn.

AB - We consider the poset of weighted partitions Πw n, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Πw n provide a generalization of the lattice Πn of partitions, which we show possesses many of the well-known properties of Πn. 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 Sn-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 Πw n has a nice factorization analogous to that of Πn.

UR - http://www.scopus.com/inward/record.url?scp=84960326039&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960326039&partnerID=8YFLogxK

U2 - 10.1090/tran/6483

DO - 10.1090/tran/6483

M3 - Article

VL - 368

SP - 6779

EP - 6818

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 10

ER -