Restricted permutations, continued fractions, and Chebyshev polynomials

الأصل · EN

Let fₙʳ(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)=∑n0 fₙʳ(k)xⁿ and F(x,y;k)=∑r0Fᵣ(x;k)yʳ. We find an explcit expression for F(x,y;k) in the form of a continued fraction. This allows us to express Fᵣ(x;k) for 1 r k via Chebyshev polynomials of the second kind.