Triangulations of cyclic polytopes and higher Bruhat orders

Recently Edelman and Reiner suggested two poset structures,S 1 ( n , d ) andS 2 ( n , d ) on the set of all triangulations of the cyclic d ‐polytope C ( n , d ) with n vertices. Both posets are generalizations of the well‐studied Tamari lattice. WhileS 2 ( n , d ) is bounded by definition, the same is not obvious forS 1 ( n , d ). In the paper by Edelman and Reiner the bounds ofS 2 ( n , d ) were also confirmed forS 1 ( n , d ) whenever d ≤5, leaving the general case as a conjecture.