2:00 pm, Seminar Hall
Mathematics/Computer Science Seminar
Tamari lattice and its extensions
LaBRI, University of Bordeaux.
The vertices of the Tamari lattice are binary trees with an order relation defined by a rotation on binary trees. Such lattice can be realized geometrically as a convex polyhedron called the associahedron. Such structure, also called Stasheff polytope, was also introduced in relation with homotopy theory. In recent years, many works have been done on the Tamari lattice.
I will give a survey of the subject: relation between the 3 structures: hypercube, associahedron and permutohedron (vertices are permutations) at the geometric, algebraic (Hopf algebras) and combinatorial level; enumeration of the intervals, which are surprisingly the same as the number of triangulations.
Tamari lattice can also be defined with Dyck paths (or ballot paths above the diagonal). In relation with the dimension of the so-called higher diagonal coinvariant. F.Bergeron introduced the m-Tamari lattice for every integer m, corresponding to paths above a line with slope 1/m. It was an open problem to extend Tamari lattice to the so-called "rational Catalan combinatorics", i.e. to paths above a line will rational slope a/b. I will finish the talk by defining extensions of Tamari lattice, far more general that just paths above the line a/b (join work with L.-F. PrÃ©ville-Ratelle).This work involves algorithmic bijections between binary trees and pair of paths (or staircase polygons).