Enumeration of non-crossing pairings on bit strings
Abstract
A non-crossing pairing on a bitstring matches 1s and 0s in a manner such that the pairing diagram is nonintersecting. By considering such pairings on arbitrary bitstrings 1n1 0m1 ... 1nr 0mr, we generalize classical problems from the theory of Catalan structures. In particular, it is very difficult to find useful explicit formulas for the enumeration function φ(n1, m1, ..., nr, mr), which counts the number of pairings as a function of the underlying bitstring. We determine explicit formulas for φ, and also prove general upper bounds in terms of Fuss-Catalan numbers by relating non-crossing pairings to other generalized Catalan structures (that are in some sense more natural). This enumeration problem arises in the theory of random matrices and free probability.