The lattice of non-crossing partitions is a partial order on objects counted by the Catalan numbers. This poset has various generalizations and interesting enumerative properties. For instance, the number of maximal chains in the poset is counted by parking functions, and its Möbius function is up to sign also a Catalan number. In this talk, we consider the generalization of the non-crossing partition lattice given by the core label order of the $\nu$-Tamari lattice, and extend classical enumerative results in this more general context.