A Conjecture on descents, inversions, and the weak order
We discuss the notion of partitions of elements in an arbitrary Coxeter system: a partition of an element w is the set of elements such that the left descents of w are the disjoint union of the left descents of the elements in the partition. This is related to the weak order of a Coxeter system and leads to an interesting conjecture that the number of right descents in w is the sum of the numbers of descents in the partition.