321-Polygon-avoiding permutations and Chebyshev polynomials

A 321-k-gon-avoiding permutation π avoids 321 and the following four patterns: k(k + 2)(k +3)⋯(2k - 1)1(2k)23⋯(k - 1)(k + 1), k(k + 2)(k +3)⋯(2k - 1)(2k)12⋯(k - 1)(k + 1), (k + 1)(k + 2)(k + 3)⋯(2k - 1)1(2k)23⋯k, (k + 1)(k + 2)(k + 3)⋯(2k - 1)(2k)123⋯k. The 321-4-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW] as a class of elements of the symmetric group whose Kazhdan-Lusztig, Poincaré polynomials, and the singular loci of whose Schubert varieties have fairly simple formulas and descriptions. Stankova and West [SW1] gave an exact enumeration in terms of linear recurrences with constant coefficients for the cases k = 2, 3, 4. In this paper, we extend these results by finding an explicit expression for the generating function for the number of 321-k-gon-avoiding permutations on n letters. The generating function is expressed via Chebyshev polynomials of the second kind.