Restricted permutations, continued fractions, and Chebyshev polynomials
T. Mansour, A. Vainshtein
Abstract
Let fnr(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12... k, and let Fr(x;k) and F(x,y;k) be the generating functions defined by Fr(x;k)=Σn0 fnr(k)xn and F(x,y;k)=Σr0Fr(x;k)yr. We find an explcit 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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