• 1 Combinatoire, théorie des nombres

We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$_$n-1$, $B$_$n$, $Ã$_$n$, and the flag weak order on the wreath product ℤ_$r$ ≀ $S$_$n$ introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$_$n-1$ case, in which case we obtain the classical Stanley symmetric function.