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In Artin-Tits groups attached to Coxeter groups of spherical type, we give acombinatorial formula to express the simple elements of the dual braid monoidsin the classical Artin generators. Every simple dual braid is obtained bylifting an $S$-reduced expression of its image in the Coxeter group, in a waywhich involves Reading's $c$-sortable elements. It has as an immediateconsequence that simple dual braids are Mikado braids (the known proofs of thisresult either require topological realizations of the Artin groups orcategorification techniques), and hence that their images in the Iwahori-Heckealgebras have positivity properties. In the classical types, this requires togive an explicit description of the inverse of Reading's bijection from$c$-sortable elements to noncrossing partitions of a Coxeter element $c$, whichmight be of independent interest. The bijections are described in terms of thenoncrossing partition models in these types. While the proof of the formula iscase-by-case, it is entirely combinatorial and we develop an approach whichreduces a uniform proof to uniformly proving a lemma about inversion sets of$c$-sortable elements.

Dual Garside structures and Coxeter sortable elements