Grothendieck Bialgebras, Partition Lattices, and Symmetric Functions in Noncommutative Variables

@article{Bergeron2006GrothendieckBP,
  title={Grothendieck Bialgebras, Partition Lattices, and Symmetric Functions in Noncommutative Variables},
  author={N. Bergeron and Christophe Hohlweg and M. Rosas and M. Zabrocki},
  journal={Electron. J. Comb.},
  year={2006},
  volume={13}
}
We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra. 

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Partitions, Rooks, and Symmetric Functions in Noncommuting Variables
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A Bijection between Atomic Partitions and Unsplitable Partitions
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