Symmetric pattern-avoiding permutations are restricted permutations which are invariant under actions of certain subgroups of D 4 , the symmetry group of a square. We examine pattern-avoiding permutations with 180 • rotational-symmetry. In particular, we use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one pattern of length four. Our results involve well-known sequences such as the alternating Fibonacci numbers, Catalan numbers, triangular numbers, and powers of two.
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