Restricted permutations, continued fractions, and Chebyshev polynomials

Details Restricted permutations, continued fractions, and Chebyshev polynomials Restricted permutations, continued fractions, and Chebyshev polynomials

1999-12-06

arxiv.org/abs/math/9912052v2

Mathematics

Combinatorics

Scientific paper

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 $F_r(x;k)=\sum_{n\gs0} f_n^r(k)x^n$ and $F(x,y;k)=\sum_{r\gs0}F_r(x;k)y^r$. We find an explcit expression for F(x,y;k) in the form of a continued fraction. This allows us to express F_r(x;k) for $1\ls r\ls k$ via Chebyshev polynomials of the second kind.

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