Counting smaller trees in the Tamari order

Combinatoire alg\'ebrique li\'ee aux ordres sur les permutations

This thesis comes within the scope of algebraic combinatorics and studies problems related to three orders on permutations: the two said weak orders (right and left) and the strong order or Bruhat order. We first look at the action of the symmetric group on multivariate polynomials. By using the divided differences operators, one can obtain some generalisations of the Schur function and form ba...

Counting smaller elements in the Tamari and m-Tamari lattices

A sequent calculus for the Tamari order

We introduce a sequent calculus with a simple restriction of Lambek’s product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, tree rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the foll...

Extending the Tamari Lattice to Some Compositions of Species

COMPOSITIONS OF SPECIES STEFAN FORCEY Abstract. An extension of the Tamari lattice to the multiplihedra is discussed, along with projections to the composihedra and the Boolean lattice. The multiplihedra and composihedra are sequences of polytopes that arose in algebraic topology and category theory. Here we describe them in terms of the composition of combinatorial species. We de ne lattice st...

A Motzkin filter in the Tamari lattice

The Tamari lattice of order n can be defined on the set Tn of binary trees endowed with the partial order relation induced by the well-known rotation transformation. In this paper, we restrict our attention to the subset Mn of Motzkin trees. This set appears as a filter of the Tamari lattice. We prove that its diameter is 2n − 5 and that its radius is n − 2. Enumeration results are given for jo...