The lattice Tamari(v) is a maule

First I will introduce a new family of posets called "maule" and recall the definition of the classical Tamari lattice defined by a "rotation" on binary trees. By translating the rotation of binary trees in the context of Dyck paths, the Tamari lattice has been extended by F. Bergeron to m-Tamari (m integer) in relation with diagonal coinvariant spaces of the symmetric group. The problem of extending this lattice to any rational number m was solved by L.-F. Préville-Ratelle and the speaker. In fact, we defined a much more general extension: for any path v with elementary steps East and North we defined a lattice Tamari(v). I will prove that this lattice Tamari(v) is also a maule, which gives a new and more simple definition of this lattice (and thus also of the usual Tamari lattice). Again, as in the first lecture seminar about maules, Catalan alternative tableaux related to the TASEP model in physics play a crucial role, but with totally different bijections. 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 theclassical associahedron for the usual Tamari lattice.