Factorizations of the canonical full cycle and symmetric q,t-polynomials

The aim of our study is to examine the relationship between minimal factorizations of the canonical full cycle that arise in lattices of non-crossing partitions, as well as the statistics attached to these objects. We shall first study properties of these factorizations that will help us establish statistic-preserving bijections among these objects. We also study some special subsets of these factorizations that come from their relationship with parking functions. Furthermore, some of the statistics on these objects give rise to interesting symmetric q,t-polynomials that we also examine.