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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the (co)homology of the poset of weighted partitions
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by Rafael S. González D’León and Michelle L. Wachs PDF
Trans. Amer. Math. Soc. 368 (2016), 6779-6818 Request permission

Abstract:

We consider the poset of weighted partitions $\Pi _n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $\Pi _n^w$ provide a generalization of the lattice $\Pi _n$ of partitions, which we show possesses many of the well-known properties of $\Pi _n$. In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on node set $\{1,2,\dots ,n\}$ having a fixed number of descents, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted $\mathfrak {S}_n$-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 $\Pi _n^w$ has a nice factorization analogous to that of $\Pi _n$.
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Additional Information
  • Rafael S. González D’León
  • Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
  • Email: dleon@math.miami.edu
  • Michelle L. Wachs
  • Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
  • MR Author ID: 179695
  • Email: wachs@math.miami.edu
  • Received by editor(s): December 13, 2013
  • Received by editor(s) in revised form: April 13, 2014, and May 12, 2014
  • Published electronically: February 2, 2016
  • Additional Notes: The first author was supported by NSF grant DMS 1202755
    The work of the second author was partially supported by a grant from the Simons Foundation (#267236) and by NSF grants DMS 0902323 and DMS 1202755.
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 6779-6818
  • MSC (2010): Primary 05E45; Secondary 05E15, 05E18, 06A11, 17B01
  • DOI: https://doi.org/10.1090/tran/6483
  • MathSciNet review: 3471077