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The enumeration of generalized Tamari intervals


Authors: Louis-François Préville-Ratelle and Xavier Viennot
Journal: Trans. Amer. Math. Soc. 369 (2017), 5219-5239
MSC (2010): Primary 06A07
DOI: https://doi.org/10.1090/tran/7004
Published electronically: March 17, 2017
MathSciNet review: 3632566
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Abstract: For any finite path $ v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$ (v)$ that consists of all the paths weakly above $ v$ with the same number of north and east steps as $ v$. For particular choices of $ v$, we recover the traditional Tamari lattice and the $ m$-Tamari lattice.

Let $ \overleftarrow {v}$ be the path obtained from $ v$ by reading the unit steps of $ v$ in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam$ (v)$ is isomorphic to the dual of the poset Tam $ (\overleftarrow {v})$. We do so by showing bijectively that the poset Tam$ (v)$ is isomorphic to the poset based on rotation of full binary trees with the fixed canopy $ v$, from which the duality follows easily. This also shows that Tam$ (v)$ is a lattice for any path $ v$. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height $ n$, can be partitioned into the (smaller) lattices Tam$ (v)$, where the $ v$ are all the paths on the square grid that consist of $ n-1$ unit steps.

We explain possible connections between the poset Tam$ (v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.


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Additional Information

Louis-François Préville-Ratelle
Affiliation: Instituto de Mathemática y Física, Universidad de Talca, 2 norte 685, Talca, Chile
Email: preville-ratelle@inst-mat.utalca.cl

Xavier Viennot
Affiliation: CNRS, LABRI, Université de Bordeaux, Bordeaux, France
Email: viennot@xavierviennot.org

DOI: https://doi.org/10.1090/tran/7004
Received by editor(s): June 17, 2014
Received by editor(s) in revised form: August 22, 2015, August 27, 2015, and June 20, 2016
Published electronically: March 17, 2017
Additional Notes: The first author was supported by the government of Chile under Proyecto Fondecyt 3140298.
Article copyright: © Copyright 2017 American Mathematical Society