Continued fractions and generalized patterns
| dc.creator | Mansour, T. | |
| dc.date | 2001-10-03 | |
| dc.date | 2001-10-20 | |
| dc.date.accessioned | 2026-07-07T04:43:38Z | |
| dc.date.available | 2026-07-07T04:43:38Z | |
| dc.description | 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 $f_{τ;r}(n)$ be the number of $1\mn3\mn2$-avoiding permutations on $n$ letters that contain exactly $r$ occurrences of $τ$, where $τ$ a generalized pattern on $k$ letters. Let $F_{τ;r}(x)$ and $F_τ(x,y)$ be the generating functions defined by $F_{τ;r}(x)=\sum_{n\geq0} f_{τ;r}(n)x^n$ and $F_τ(x,y)=\sum_{r\geq0}F_{τ;r}(x)y^r$. We find an explicit expression for $F_τ(x,y)$ in the form of a continued fraction for where $τ$ given as a generalized pattern; $τ=12\mn3\mn...\mn k$, $τ=21\mn3\mn...\mn k$, $τ=123... k$, or $τ=k... 321$. In particularly, we find $F_τ(x,y)$ for any $τ$ generalized pattern of length 3. This allows us to express $F_{τ;r}(x)$ via Chebyshev polynomials of the second kind, and continued fractions. | |
| dc.description | 16 pages | |
| dc.identifier | https://arxiv.org/abs/math/0110037 | |
| dc.identifier | http://arxiv.org/abs/math/0110037 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62311 | |
| dc.subject | Combinatorics | |
| dc.title | Continued fractions and generalized patterns | |
| dc.type | text |