An equivalence of multistatistics on permutations

We prove a conjecture of J.-C. Novelli, J.-Y. Thibon, and L.K. Williams (2010) about an equivalence of two triples of statistics on permutations. To prove this conjecture, we build a bijection through different combinatorial objects, starting with a Catalan-based object related to the PASEP. As a byproduct of this research, we also provide a new co-sylvester class-preserving bijection on permutations.