Tableaux and chains in a new partial order of Sn

AbstractWe define a new poset on the symmetric group Sn. It is subposet of the weak ordering of Sn with the property that every interval is a distributive lattice. For each principal ideal we explicitly compute the poset of join-irreducibles and use this to get an expression for the number of maximal chains in these intervals. In the case of certain principal ideals the number of maximal chains is given by the number of shifted tableaux of a certain shape. In particular, in the principal ideal generated by the reverse permutation the number of maximal chains is given by the number of shifted tableaux of staircase shape. We relate this to similar results for the weak ordering and show how bijections that work in that case restrict to this new poset. This new poset arises naturally from the study of convex geometries. We detail these connections and give an alternative definition for this poset in this context