Continued fractions and generalized patterns

Babson and Steingrimsson (2000, Séminaire Lotharingien de Combinatoire, B44b, 18) 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 1-3-2-avoiding permutations on n letters that contain exactly r occurrences of τ, where τ is 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) = ∑_n≥0f_τ;r(n)x ⁿ and F_τ(x, y) = ∑_r≥0 F _τ;r(x)y^r. We find an explicit expression for F _τ(x, y) in the form of a continued fraction for τ given as a generalized pattern: τ = 12-3-...-k, τ = 21-3-...-k, τ = 123...k, or τ = k...321. In particular, 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.