Continued fractions and generalized patterns

Details Continued fractions and generalized patterns Continued fractions and generalized patterns

2001-10-03

arxiv.org/abs/math/0110037v2

Mathematics

Combinatorics

16 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Continued fractions and generalized patterns does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Continued fractions and generalized patterns, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Continued fractions and generalized patterns will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-475092