Generating trees and the Catalan and Schro¨der numbers
Discrete Mathematics
Regular Article: The Enumeration of Permutations with a Prescribed Number of 驴Forbidden驴 Patterns
Advances in Applied Mathematics
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
Forbidden subsequences and Chebyshev polynomials
Discrete Mathematics - Special issue on selected papers in honor of Henry W. Gould
Permutations with Restricted Patterns and Dyck Paths
Advances in Applied Mathematics
Horse paths, restricted 132-avoiding permutations, continued fractions, and Chebyshev polynomials
Discrete Applied Mathematics
Horse paths, restricted 132-avoiding permutations, continued fractions, and Chebyshev polynomials
Discrete Applied Mathematics
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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 Fτr (x) = Σn≥0 fτ;r (n)xn and Fτ(x, y) = Σr≥0 Fτ;r(x)yr. 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.