Letters of a given size in Catalan words

A Catalan word with n letters, w = w1w2 . . .wn over the set of positive integers is a word in which w1 = 1 and for each i ∈ {2, 3 . . ., n}, the inequality wi ≤ wi-1 + 1 holds.We find generating functions A(x, qi) for the number of letters equal to i in Catalan words with n letters.We also develop the generating function A(x, qi|j) that tracks the number of letters i in Catalan words ending in a letter j.The relationship between A(x, qi) and A(x, qi|j) is fundamental to the kernel method used to obtain these generating functions.