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

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


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.).

Original languageEnglish
Pages (from-to)1568-1582
Number of pages15
JournalJournal of Difference Equations and Applications
Issue number11
StatePublished - 26 Nov 2014

Bibliographical note

Publisher Copyright:
2014, © 2014 Taylor & Francis.


  • Chebyshev polynomials
  • combinatorial statistics
  • functional equation
  • recurrence relation

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics


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