Continued fractions, statistics, and generalized patterns

Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following \cite{BCS}, let $e_k\pi$ (respectively; $f_k\pi$) be the number of the occurrences of the generalized pattern $12\text{mn}3\text{mn}\mathrm{.}\mathrm{.}\mathrm{.}\text{mn}k$ (respectively; $21\text{mn}3\text{mn}\mathrm{.}\mathrm{.}\mathrm{.}\text{mn}k$) in $\pi$. In the present note, we study the distribution of the statistics $e_k\pi$ and $f_k\pi$ in a permutation avoiding the classical pattern $1\text{mn}3\text{mn}2$. Also we present an applications, which relates the Narayana numbers, Catalan numbers, and increasing subsequences, to permutations avoiding the classical pattern $1\text{mn}3\text{mn}2$ according to a given statistics on $e_k\pi$, or on $f_k\pi$.