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 fτ ;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 Fτ ;r (x) and Fτ (x, y) be the generating functions defined by Fτ ;r (x) = ∑ n≥0 fτ ;r (n)x n and Fτ (x, y) = ∑ r≥0 Fτ ;r (x)y r . We find an explicit expression for Fτ (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 Fτ (x, y) for any τ generalized pattern of length 3. This allows us to express Fτ ;r (x) via Chebyshev polynomials of the second kind and continued fractions.