Staircase words and chebyshev polynomials

Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger

Research output: Contribution to journalArticlepeer-review


A word σ = σx · · · σnover the alphabet [k] = {1, 2,..., k} is said to be a staircase if there are no two adjacent letters with difference greater than 1. A word σ is said to be staircase-cyclic if it is a staircase word and in addition satisfies |σn - σ| ≤ 1. We find the explicit generating functions for the number of staircase words and staircase-cyclic words in [k]n, in terms of Chebyshev polynomials of the second kind. Additionally, we find explicit formulæ for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate staircase necklaces, which are staircase-cyclic words that are not equivalent up to rotation.

Original languageEnglish
Pages (from-to)81-95
Number of pages15
JournalApplicable Analysis and Discrete Mathematics
Issue number1
StatePublished - Apr 2010
Externally publishedYes


  • Chebyshev polynomial
  • Generating function
  • Staircase cyclic word
  • Staircase necklace
  • Staircase word
  • Tridiagonal matrix

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


Dive into the research topics of 'Staircase words and chebyshev polynomials'. Together they form a unique fingerprint.

Cite this