A Generic Numbering System based on Catalan Families of Combinatorial Objects

We study novel arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects. Our algorithms work on a generic representation that we illustrate on instances like ordered binary and multiway trees, balanced parentheses languages as well as the usual bitstring-based natural numbers seen through the same generic interface as members of the Catalan family. For numbers corresponding to Catalan objects of low representation complexity, our algorithms provide super-exponential gains while their average and worst case complexity is within constant factors of their traditional counterparts.