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Continued Fractions and Generalized Patterns

✍ Scribed by Toufik Mansour

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 140 KB
Volume: 23
Category: Article
ISSN: 0195-6698
DOI: 10.1006/eujc.2002.0566

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✦ Synopsis

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≥0 f 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:

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.

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