Restricted Permutations, Continued Fractions, and Chebyshev Polynomials

Let fr n(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12 . . . k, and let Fr(x; k) and F (x, y; k) be the generating functions defined by Fr(x; k) = Pn 0 fr n(k)xn and F (x, y; k) = Pr 0 Fr(x; k)yr. We find an explicit expression for F (x, y; k) in the form of a continued fraction. This allows us to express Fr(x; k) for 1 6 r 6 k via Chebyshev polynomials of the second kind. 2000 Mathematics Subject Classification: Primary 05A05, 05A15; Secondary 30B70, 42C05 Typeset by AMS-TEX 1 

• Type: Journal
• Year: 2000
• Where: COMBINATORICS
• Authors: Toufik Mansour, Alek Vainshtein
• Keywords: 132-avoiding Permutations, COMBINATORICS 2000, Explicit Expression, Generating Functions