The F–M triangle transformation conjecture for crystallographic root systems

Let $W$ be the Weyl group of a crystallographic root system $\Phi$ of rank $n$, and let $F(x,y)$ and $M(x,y)$ denote respectively the $F$-triangle for $\Phi$ and the $M$-triangle for $W$, where

$$M(x,y)=\sum_{a \leq b} \mu(a,b)x^{\operatorname{rk}(b)}y^{\operatorname{rk}(a)}.$$

F–M triangle transformation conjecture. The two generating functions are related by the invertible transformation

$$(1-y)^n F\left(\frac{x+y}{1-y},\frac{y}{1-y}\right)=M\left(-x,-\frac{y}{x}\right).$$

This conjecture proposes a uniform relation between the face-enumerating $F$-triangle of the generalized associahedron and the interval Möbius-number $M$-triangle of the associated noncrossing partition lattice. The supplied text gives no resolution status, so the conjecture is recorded as open.