Staircase words and chebyshev polynomials

Arnold Knopfmacher; Toufik Mansour; Augustine Munagi; Helmut Prodinger

doi:10.2298/AADM1000010K

Staircase words and chebyshev polynomials

Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger

Research output: Contribution to journal › Article › peer-review

Abstract

A word σ = σ_x · · · σ_nover 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 |σ_n - 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 language	English
Pages (from-to)	81-95
Number of pages	15
Journal	Applicable Analysis and Discrete Mathematics
Volume	4
Issue number	1
DOIs	https://doi.org/10.2298/AADM1000010K
State	Published - Apr 2010
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.2298/AADM1000010K

Cite this

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AU - Knopfmacher, Arnold

AU - Mansour, Toufik

AU - Munagi, Augustine

AU - Prodinger, Helmut

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N2 - 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.

AB - 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.

KW - Chebyshev polynomial

KW - Generating function

KW - Staircase cyclic word

KW - Staircase necklace

KW - Staircase word

KW - Tridiagonal matrix

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Staircase words and chebyshev polynomials

Abstract

Keywords

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