Part II: Sublattices of Associahedra and Permutohedra Equations satisfied by Associahedra

The set of permutations on n-elements is a poset when endowed with weak Bruhat ordering. Actually, this poset is a lattice since each finite subset of permutations has both a supremum and infimum. Such a lattice is known as the Permutohedron on n-elements. A strictly related lattice is Associahedron on n+1 letters. Its elements are all the ways of parenthesizing a word of length n+1 (i.e. binary trees with n+1 leaves). The order is the transitive closure of the operation that replaces a sub-parenthesized word (uv)w with u(vw). These talks shall describe undergoing work on Associahedra and Permutohedra. It was conjectured in 1992 by Geyer that every bounded image of a free lattice embeds into an Associahedron. We disprove the conjecture, and go further to state an analogous statement for the Permutohedra: not every bounded image of a free lattice embeds into a Permutohedron. While the first result leads to discover non-trivial lattice-theoretic identities holding in all the Associahedra, we do not yet know any non-trivial identity that holds in all the Permutohedra. I shall present these results and the open research directions. A particular emphasis shall be given to the tools developed to obtain the results, including the OD-graph of a lattice, polarized measures, maximal subdirect decompositions, splitting equations, etc. Joint work with Fred Wehrung.