Abstract: The weak order and Bruhat order are well-known posets on permutations, the weak order being a lattice which is contained in the Bruhat order. Recently a new distributive lattice on permutations called „middle order“ was discovered, containing the weak order and contained in the Bruhat order. We generalize this result by constructing C_{n-1} distributive lattices on permutations of size n with the same property, using a simple bijection between permutations and ideals of posets. We then show these lattices are the only distributive lattices between weak and Bruhat orders, and we also consider generalizations of middle orders in other Coxeter groups.

Abstract: Gog and Magog triangles are simple combinatorial objects which are equienumerated. Howewer, the problem of finding an explicit bijection between these has been an open problem since the 80’s. These are related to other interesting objects such as alternating sign matrices, plane partitions or aztec diamond tillings. All these objects can be ordered in such a way that the obtained posets are distributive lattices. We will present Gog and Magog triangles under a lattice-theoretic point of view, giving new explanations of the link between alternating sign matrices and aztec diamond tillings, or between the lattice of Gog triangles, the Bruhat and weak orders on permutations.