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Published in:
2012 | OriginalPaper | Chapter
Catalan Lattices on Series Parallel Interval Orders
Authors : Filippo Disanto, Luca Ferrari, Renzo Pinzani, Simone Rinaldi
Published in: Associahedra, Tamari Lattices and Related Structures
Publisher: Springer Basel
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Using the notion of series parallel interval order, we propose a unified setting to describe Dyck lattices and Tamari lattices (two well-known lattice structures on Catalan objects) in terms of basic notions of the theory of posets. As a consequence of our approach, we find an extremely simple proof of the fact that the Dyck order is a refinement of the Tamari one. Moreover, we provide a description of both the weak and the strong Bruhat order on 312-avoiding permutations, by recovering the proof of the fact that they are isomorphic to the Tamari and the Dyck order, respectively; our proof, which simplifies the existing ones, relies on our results on series parallel interval orders.