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

Pages (from-to) | 1029-1040 |

Number of pages | 12 |

Journal | Discrete Mathematics and Theoretical Computer Science |

State | Published - 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

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

**On the poset of weighted partitions.** / D'León, Rafael S González; Galloway, Michelle L.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - On the poset of weighted partitions

AU - D'León, Rafael S González

AU - Galloway, Michelle L

PY - 2013

Y1 - 2013

N2 - 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 On-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 nhas a nice factorization analogous to that of ∏n.

AB - 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 On-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 nhas a nice factorization analogous to that of ∏n.

KW - Free lie algebra

KW - Partitions

KW - Poset topology

KW - Rooted trees

UR - http://www.scopus.com/inward/record.url?scp=84887500249&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887500249&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84887500249

SP - 1029

EP - 1040

JO - Discrete Mathematics and Theoretical Computer Science

JF - Discrete Mathematics and Theoretical Computer Science

SN - 1365-8050

ER -