Continued fractions and generalized patterns

Abstract

In [BS] Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let $f_{\tau;r}(n)$ be the number of $1\mn3\mn2$-avoiding permutations on $n$ letters that contain exactly $r$ occurrences of $\tau$, where $\tau$ a generalized pattern on $k$ letters. Let $F_{\tau;r}(x)$ and $F_\tau(x,y)$ be the generating functions defined by $F_{\tau;r}(x)=\sum_{n\geq0} f_{\tau;r}(n)x^n$ and $F_\tau(x,y)=\sum_{r\geq0}F_{\tau;r}(x)y^r$. We find an explicit expression for $F_\tau(x,y)$ in the form of a continued fraction for where $\tau$ given as a generalized pattern; $\tau=12\mn3\mn...\mn k$, $\tau=21\mn3\mn...\mn k$, $\tau=123... k$, or $\tau=k... 321$. In particularly, we find $F_\tau(x,y)$ for any $\tau$ generalized pattern of length 3. This allows us to express $F_{\tau;r}(x)$ via Chebyshev polynomials of the second kind, and continued fractions.