Restricted permutations, continued fractions, and Chebyshev polynomials

Toufik Mansour; Alek Vainshtein

doi:10.37236/1495

Restricted permutations, continued fractions, and Chebyshev polynomials

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

Abstract

Let f_n^r(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12 . . . k, and let F _r(x; k) and F(x, y; k) be the generating functions defined by F _r(x; k) = Σ_n≥0 f_n^r(k)x ⁿ and F(x, y; k) = Σ_r≥0 F_r(x; k)y ^r. We find an explicit expression for F(x, y; k) in the form of a continued fraction. This allows us to express F_r(x; k) for 1 ≤ r ≤ k via Chebyshev polynomials of the second kind.

Original language	English
Pages (from-to)	1-9
Number of pages	9
Journal	Electronic Journal of Combinatorics
Volume	7
Issue number	1 R
DOIs	https://doi.org/10.37236/1495
State	Published - 2000

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

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10.37236/1495

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abstract = "Let fnr(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12 . . . k, and let F r(x; k) and F(x, y; k) be the generating functions defined by F r(x; k) = Σn≥0 fnr(k)x n and F(x, y; k) = Σr≥0 Fr(x; k)y r. We find an explicit expression for F(x, y; k) in the form of a continued fraction. This allows us to express Fr(x; k) for 1 ≤ r ≤ k via Chebyshev polynomials of the second kind.",

author = "Toufik Mansour and Alek Vainshtein",

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AU - Vainshtein, Alek

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AB - Let fnr(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12 . . . k, and let F r(x; k) and F(x, y; k) be the generating functions defined by F r(x; k) = Σn≥0 fnr(k)x n and F(x, y; k) = Σr≥0 Fr(x; k)y r. We find an explicit expression for F(x, y; k) in the form of a continued fraction. This allows us to express Fr(x; k) for 1 ≤ r ≤ k via Chebyshev polynomials of the second kind.

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Restricted permutations, continued fractions, and Chebyshev polynomials

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