Tamari lattices, forests and Thompson monoids

Tamari Lattices and the symmetric Thompson monoid

We investigate the connection between Tamari lattices and the Thompson group F , summarized in the fact that F is a group of fractions for a certain monoid F sym whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice operations are the counterparts of the least common multiple and greatest common divisor operations in F sym. As an application, we show tha...

Chains of Maximum Length in the Tamari Lattice

The Tamari lattice Tn was originally defined on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Although in several related lattices, the number of maximal chains is known, quoting Knuth, “The enumeration of such paths in Tamari lattices remains mysterious.” The lengths of maximal chains vary over a great range. In this paper, we focu...

Counting smaller elements in the Tamari and m-Tamari lattices

Geometry of ν-Tamari lattices in types A and B

In this extended abstract, we exploit the combinatorics and geometry of triangulations of products of simplices to reinterpret and generalize a number of constructions in Catalan combinatorics. In our framework, the main role of “Catalan objects” is played by (I, J)-trees: bipartite trees associated to a pair (I, J) of finite index sets that stand in simple bijection with lattice paths weakly a...

Monoid generalizations of the Richard Thompson groups

The groups Gk,1 of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call Mk,1, and to inverse monoids, called Invk,1; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids Mk,1 have connections with circuit complexity (studied in another paper). Here we prove that Mk,1 and Invk,1 are congruence-s...