Catalan lattices on series parallel interval orders

Details Catalan lattices on series parallel interval orders Catalan lattices on series parallel interval orders

2010-07-12

arxiv.org/abs/1007.1911v1

Mathematics

Combinatorics

16 pages, 10 figures, submitted

Scientific paper

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.

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