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.

Catalan lattices on series parallel interval orders / F. Disanto; L. Ferrari; R. Pinzani; S. Rinaldi. - STAMPA. - (2012), pp. 323-338. [10.1007/978-3-0348-0405-9_16]

Catalan lattices on series parallel interval orders

FERRARI, LUCA;PINZANI, RENZO;
2012

Abstract

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.
2012
9783034804042
Associahedra, Tamari Lattices, and Related Structures
323
338
F. Disanto; L. Ferrari; R. Pinzani; S. Rinaldi
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/593132
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