Lobb's generalisation of Catalan's parenthesisation problem revisited

Abstract

In 1838, the Belgian mathematician Eugene C. Catalan (1814-1894) discovered that the number Cn of well-fonned sequences, with n pairs of left and right parentheses, is given by where n > 0 [1, 2]. For example, there are exactly five well-formed sequences with three pairs of left and right parentheses: ()()(), ()(()), (())(), (()()), ((())). The case n = 0 yields the null sequence, often denoted by λ. Notice that ()) and ((()()), for example, are not correctly parenthesised.