The set of permutations on nelements 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 nelements. 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 subparenthesized word (uv)w with u(vw). In this talk I 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 nontrivial latticetheoretic identities holding in all the Associahedra, we do not yet know any nontrivial identity that holds in all the Permutohedra. Joint work with Fred Wehrung.
