Restricted permutations, continued fractions, and Chebyshev polynomials

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 $F_r(x;k)={\sum }_{n\text{gs}0}{f}_n^r(k)x^n$ and $F(x,y;k)={\sum }_{r\text{gs}0}{F}_r(x;k)y^r$. 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 $1\text{ls}r\text{ls}k$ via Chebyshev polynomials of the second kind.