ENUMERATION OF CATALAN AND SMOOTH WORDS ACCORDING TO CAPACITY

A bargraph is a sequence of rectangles lying in the first quadrant each of unit width and positive integral length whose widths are flush with the x-axis starting from the origin. By the capacity of a bargraph, we mean the amount of a liquid that would be retained when poured over the bargraph from above by virtue of its shape. In this paper, we consider the capacity statistic on two classes of words satisfying certain growth restrictions, represented geometrically as bargraphs. A Catalan word w = w₁ · · · w_n is one with positive integer entries such that w_i+1 − w_i ≤ 1 for 1 ≤ i ≤ n − 1, with w₁ = 1. Let A_n denote the set of Catalan words of length n, which are enumerated by the n-th Catalan number C_n for all n ≥ 1. We derive an explicit formula for the generating function of the capacity distribution on A_n and study further properties of this distribution such as the number of members of A_n achieving the maximum and minimum capacity. A similar treatment is provided for the set of smooth words, that is, those satisfying the condition |w_i+1 − w_i | ≤ 1 for 1 ≤ i ≤ n − 1 instead, and also study the distribution of capacity on a restricted class of smooth words. As special cases of our results in the various cases, we obtain infinite series identities involving the reciprocals of Chebyshev polynomials.