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Weighted partitions

Abstract : In this extended abstract we consider the poset of weighted partitions Π _n^w, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π _n^w 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 compute the Möbius invariant in terms of rooted trees, 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 _n^w has a nice factorization analogous to that of Π _n.
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https://hal.inria.fr/hal-01229690
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Submitted on : Tuesday, November 17, 2015 - 10:20:00 AM
Last modification on : Tuesday, March 7, 2017 - 3:23:22 PM
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Rafael González S. d'León, Michelle L. Wachs. Weighted partitions. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.1029-1040. ⟨hal-01229690⟩

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