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

Abstract

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 1 □ 23 (there is no occurrence π i < π j < π j + 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 1 □ 23 , and certain additional patterns. We also give generating functions for permutations avoiding 132 and 1 □ 23 and containing certain additional patterns exactly once. In all cases we express these generating functions in terms of Chebyshev polynomials of the second kind.