Enumeration of non-crossing pairings on bit strings

A non-crossing pairing on a bit string is a matching of 1s and 0s in the string with the property that the pairing diagram has no crossings. For an arbitrary bit-string w=p₁¹q₁⁰…pr¹qr⁰, let φ(w) be the number of such pairings. This enumeration problem arises when calculating moments in the theory of random matrices and free probability, and we are interested in determining useful formulas and asymptotic estimates for φ(w). Our main results include explicit formulas in the “symmetric” case where each pi=qi, as well as upper and lower bounds for φ(w) that are uniform across all words of fixed length and fixed r. In addition, we offer more refined conjectural expressions for the upper bounds. Our proofs follow from the construction of combinatorial mappings from the set of non-crossing pairings into certain generalized “Catalan” structures that include labeled trees and lattice paths.