TY - JOUR

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

AU - González D’león, Rafael S.

AU - Wachs, Michelle L.

N1 - Publisher Copyright:
© 2016 American Mathematical Society.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - We consider the poset of weighted partitions Πwn, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Πwn 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 Πwn has a nice factorization analogous to that of Πn.

AB - We consider the poset of weighted partitions Πwn, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Πwn 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 Πwn has a nice factorization analogous to that of Πn.

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U2 - 10.1090/tran/6483

DO - 10.1090/tran/6483

M3 - Article

AN - SCOPUS:84960326039

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 -