  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 construct 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.
