On pattern avoiding alternating permutations

An alternating permutation of length nn is a permutation π=π1π2⋯πnπ=π1π2⋯πn such that π1<π2>π3<π4>⋯π1<π2>π3<π4>⋯. Let AnAn denote the set of alternating permutations of {1,2,…,n}{1,2,…,n}, and let An(σ)An(σ) be the set of alternating permutations in AnAn that avoid a pattern σσ. Recently, Lewis used generating trees to enumerate A2n(1234)A2n(1234), A2n(2143)A2n(2143) and A2n+1(2143)A2n+1(2143), and he posed some conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns of length four. Some of these conjectures have been proved by Bóna, Xu and Yan. In this paper, we prove two conjectured relations |A2n+1(1243)|=|A2n+1(2143)||A2n+1(1243)|=|A2n+1(2143)| and |A2n(4312)|=|A2n(1234)||A2n(4312)|=|A2n(1234)|.