  Several authors have examined connections between permutations which avoid
132, continued fractions, and Chebyshev polynomials of the second kind. In this
paper we prove analogues of some of these results for permutations which avoid
1243 and 2143. Using tools developed to prove these analogues, we give
enumerations and generating functions for permutations which avoid 1243, 2143,
and certain additional patterns. We also give generating functions for
permutations which avoid 1243 and 2143 and contain certain additional patterns
exactly once. In all cases we express these generating functions in terms of
Chebyshev polynomials of the second kind.
