Continued Fractions, Statistics, And Generalized Patterns

Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following [BCS], let ekπ (respectively; fkπ) be the number of the occurrences of the generalized pattern 12-3. . . -k (respectively; 21-3. . . -k) in π. In the present note, we study the distribution of the statistics ekπ and fkπ in a permutation avoiding the classical pattern 1-3-2. Also we present an applications, which relates the Narayana numbers, Catalan numbers, and increasing subsequences, to permutations avoiding the classical pattern 1-3-2 according to a given statistics on ekπ, or on fkπ.