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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 anddoi:10.37236/1679 fatcat:yhneetptwfgtzldqtjlysbmwwi