Geodesic
Tamari Lattices for Parabolic Quotients of the Symmetric Group
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015).

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We present a generalization of the Tamari lattice to parabolic quotients of the symmetric group. More precisely, we generalize the notions of 231-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients, and show bijectively that these sets are equinumerous. Furthermore, the restriction of weak order on the parabolic quotient to the parabolic 231-avoiding permutations is a lattice quotient. Lastly, we suggest how to extend these constructions to all Coxeter groups. 
@article{DMTCS_2015_special_285_a78,
     author = {M\"uhle, Henri and Williams, Nathan},
     title = {Tamari {Lattices} for {Parabolic} {Quotients} of the {Symmetric} {Group}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)},
     year = {2015},
     doi = {10.46298/dmtcs.2534},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2534/}
}
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Mühle, Henri; Williams, Nathan. Tamari Lattices for Parabolic Quotients of the Symmetric Group. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015). doi : 10.46298/dmtcs.2534. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2534/

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