An alternating permutation of length n is a permutation π = π 1 π 2 · · · π n such that π 1 < π 2 > π 3 < π 4 > · · · . Let A n denote set of alternating permutations of {1, 2, . . . , n}, and let A n (σ) be set of alternating permutations in A n that avoid a pattern σ. Recently, Lewis used generating trees to enumerate A 2n (1234), A 2n (2143) and A 2n+1 (2143), and he posed several conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns. Some of these conjectures have been proved by Bóna, Xu and Yan. In this paper, we prove the two relations |A 2n+1 (1243)| = |A 2n+1 (2143)| and |A 2n (4312)| = |A 2n (1234)| as conjectured by Lewis.
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