Horse paths, restricted 132-avoiding permutations, continued fractions, and Chebyshev polynomials

Several authors have examined connections among 132-avoiding permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we find analogues for some of these results for permutations π avoiding 132 and 1223 (there is no occurrence πi < πj < πj+1 such that 1 ≤ i ≤ j − 2) and provide a combinatorial interpretation for such permutations in terms of lattice paths. Using tools developed to prove these analogues, we give enumerations and generating functions for permutations which avoid both 132 and 1223, and certain additional patterns. We also give generating functions for permutations avoiding 132 and 1223 and containing certain additional patterns exactly once. In all cases we express these generating functions in terms of Chebyshev polynomials of the second kind.