Restricted permutations, continued fractions, and Chebyshev polynomials

Abstract

Let $f_n^r(k)$ be the number of 132-avoiding permutations on $n$ letters that contain exactly $r$ occurrences of $12\dots k$, and let $F_r(x;k)$ and $F(x,y;k)$ be the generating functions defined by $F_r(x;k)=\sum_{n\ge 0} f_n^r(k)x^n$ and $F(x,y;k)=\sum_{r\ge 0}F_r(x;k)y^r$. 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 $1\le r\le k$ via Chebyshev polynomials of the second kind.