79th SLC (Séminaire Lotharingien de Combinatoire), Bertinoro, Italy, 11 September 2017

**Maule: tilings, Young and Tamari lattices under the same roof **

** **** **slides part I** **(pdf, 25 Mo) tilings, Young and Tamari lattices as maules

** **slides part II (pdf, 26 Mo)** ** Tamari(v) as a maule + comments and references added after the talk** **

Same lecture is given at IMSc, Chennai, India in two video-recorded Maths seminars 19 and 26 February 2018

(slightly augmented) set of slides and video links are available on this page.** **

**abstract**

We introduce a new family of posets which I propose to call "maule". Every finite subset of the square lattice generates a maule by a dynamical process of particles moving on the square lattice. Three well-known lattices are maules: Ferrers diagrams Y(λ) contained in a given diagram λ (ideal of the Young lattice), some tilings on the triangular lattice (equivalent to plane partitions) and the very classic Tamari lattice defined with the notion of "rotation" on binary trees. We thus get a new simple definition of the Tamari lattice. Curiously, the concept of alternative tableaux plays a crucial role in this work. Such tableaux were introduced in a totally different context: the very classical model called PASEP ("partially asymmetric exclusion process"), a toy model in the physics of dynamical systems. Here we need only the subclass of Catalan alternative tableaux, corresponding to the TASEP ("totally asymmetric exclusion process").

By translating the rotation of binary trees in the context of Dyck paths, the classical Tamari lattice has been extended by F.Bergeron to m-Tamari (m integer), and then to any m rational by L.-F. Préville-Ratelle and the speaker. More generally, we defined a Tamari lattice for any path v with elementary steps East end North. We prove that this lattice Tamari(v) is also a maule, which gives a new and more simple definition of this lattice. Again, alternative tableaux (in the case of Catalan) and its avatars (permutation tableaux, tree-like tableaux and staircase tableaux) play a crucial role. These tableaux allow to relate this work with the recent work of C.Ceballos, A. Padrol et C. Sarmiento giving a geometric realization of Tamari(v), analogue to the classical associahedron for the classical Tamari lattice. With this concept of maule, we can also define a new lattice YTam(λ,v), ""mixing" the Young lattice Y(λ) and the Tamari lattice Tamari(v).

ps: "maule" is a Mapuche word (pronounce « ma-ou-lé ») which is one of the areas in Chile, together with the name of a river crossing this area where this work was done, thanks to the invitation of Luc Lapointe from Talca university.

A Catalan alternative tableau

and its corresponding Catalan staircase alternative tableau

note added:

In May 2018, C. Ceballos, A. Padrol and C. Sarmiento published the paper arXiv: 1805:03566 [Math.CO]

"The ν-Tamari lattice as the rotation lattice of ν -trees"

Some results of their paper intersect some results given at this 79th SLC, in particular the fact that the v-Tamari lattice defined by L.-F. Préville-Ratelle and the speaker (in terms of a flip in a pair of paths) corresponds to a rotation in the associated v-tree (theorem 3.2, which is in fact the title of the paper).

This equivalence is proved by using the associated Catalan alternative tableau and v-Tamari lattice defined with Γ-moves:

p124 (first part of slides) A rotation in the binary tree (the v-tree) corresponds exactly to a certain Γ-move in the associated Catalan alternative tableau,

p52-53 (second part of slides) Equivalence between a flip defining the covering relation of Tamari(v) and a Γ-move.