A conjecture on descents, inversions and the weak order

In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system $(W,S)$: a partition of an element $w$ is a subset $\mathcal{P}\subseteq W$ such that the left inversion set of $w$ is the disjoint union of the left inversion set of the elements in $\mathcal{P}$. Partitions of elements of $W$ arises in the study of the Belkale-Kumar product on the cohomology $H^{\ast }(X,\mathbb{Z})$, where $X$ is the complete flag variety of any complex semi-simple algebraic group. Partitions of elements in the symmetric group ${\mathcal{S}}_n$ are also related to the {\em Babington-Smith model} in algebraic statistics or to the simplicial faces of the Littlewood-Richardson cone. We state the conjecture that the number of right descents of $w$ is the sum of the number of right descents of the elements of $\mathcal{P}$ and prove that this conjecture holds in the cases of symmetric groups (type $A$) and hyperoctahedral groups (type $B$).