Continued fractions and generalized patterns

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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 F τ;r(x) = ∑n≥0fτ;r(n)x n and Fτ(x, y) = ∑r≥0 F τ;r(x)yr. 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 Fτ (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.

Original languageEnglish
Pages (from-to)329-344
Number of pages16
JournalEuropean Journal of Combinatorics
Issue number3
StatePublished - Apr 2002
Externally publishedYes

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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