Enumeration of non-crossing pairings on bit strings

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 1101 . . . 1r0r , 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.