Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partititions and the Farahat-Higman algebra

We introduce a new pair of mutually dual bases of noncommutative symmetric functions and quasi-symmetric functions, and use it to derive generalizations of several results on the reduced incidence algebra of the lattice of noncrossing partitions. As a consequence, we obtain a quasi-symmetric version of the FarahatHigman algebra.