Chebyshev polynomials and statistics on a new collection of words in the Catalan family

Recently, a new class of words, denoted by (Formula presented.) , was shown to be in bijection with a subset of the Dyck paths of length (Formula presented.) having cardinality the Catalan number (Formula presented.). Here, we consider statistics on (Formula presented.) recording the number of occurrences of a letter (Formula presented.). In the cases (Formula presented.) and (Formula presented.) , we are able to determine explicit expressions for the number of members of (Formula presented.) containing a given number of zeros or ones, which generalizes the prior result. To do so, we make use of recurrences to derive a functional equation satisfied by the generating function, which we solve by a new method employing Chebyshev polynomials. In the case (Formula presented.) , our result is equivalent to a prior one concerning the distribution of the initial rise statistic on Dyck paths. Recurrences and generating function formulas are also provided in the case of general (Formula presented.).