321-Avoiding Permutations and Chebyshev Polynomials

Abstract

In [6] it was shown that the generating function for the number of permutations on n letters avoiding both 321 and (d + 1)(d + 2)...k12... d is given by $\frac{{2t{{U}_{{k - 1}}}\left( t \right)}}{{{{U}_{k}}\left( t \right)}}$ for all k > 2, 2 ≥ d + 1 ≥ k, where U_m is the mth Chebyshev polynomial of the second kind and $t = \frac{1} {{2\sqrt x }}.$ In this paper we present three different classes of 321-avoiding permutations which are enumerated by this generating function.