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Call a permutation alpha of the set [n] = {1, 2, ..., n} k-free if alpha contains no k-term arithmetic progression. We prove that the number of 3-free permutations of [n] is at most 2.7n 21 for all n ≥ 11. This result substantially improves the longstanding upper bound of &fll0;n+12&flr0;!&ceill0;n +12&ceilr0;! for the number of such permutations. We also prove that at least &fll0;n2&flr0; - 6 entries at each end of a 3-free permutation are of the same parity. The proof of the new upper bound is based on two critical ideas. First, that there exists a partition of the set of all 3-free permutations of [2 n] so that each cell in the partition corresponds naturally to a unique ordered pair of 3-free permutations of [n]. Second, that each cell in the partition has at most twenty members.