2014/07/14 by Toufik Mansour, Mansour, Toufik, Mark Shattuck +1
Agricultural and Biological Sciences · Computer Science · Mathematics · #05A05 #05A15 #11B37 #Advanced Combinatorial Mathematics #Botanical Research and Chemistry #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1407.3516
openalex publication_date 2014/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently, a new class of words, denoted by Ln, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on Ln recording the number of occurrences of a letter i. In the cases i = 0 and i = 1, we are able to determine explicit expressions for the number of members of Ln 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. Recurrences and generating function formulas are also provided in the case of general i.