Some Remarks on Catalan Families

of n words of length r obtained by splitting all permutations of {O, 1, ... , rn I} into n pieces of equal length r. It suffices therefore to give a bijection between the set of all n-fold products formed from (r l)n + 1 different elements XQ, Xl' ... , x(r-l)n and the set A~. To simplify the exposition we shall write i instead of Xi and consider the usual order on the integers. Let P be any such product. It is formed using n pairs of brackets. Consider all inner brackets (i.e. those which contain no other brackets) and order them with respect to increasing largest elements. Choose the first inner bracket in this ordering and call it (r l)n + 1. Then P may be written as an (n I)-fold product of elements from a subset A £: {O, 1, ... , (r l)n + I} with (r 1)(n 1) + 1 elements. Iterating this procedure we get a uniquely determined sequence of n brackets (r l)n + 1, ... , rn. This sequence defines an element of A~.