On pattern avoiding alternating permutations

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 the set of alternating permutations of { 1 , 2 , … , n } , and let A n ( σ ) be the set of alternating permutations in A n that avoid a pattern σ . Recently, Lewis used generating trees to enumerate A 2 n ( 1234 ) , A 2 n ( 2143 ) and A 2 n + 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 | A 2 n + 1 ( 1243 ) | = | A 2 n + 1 ( 2143 ) | and | A 2 n ( 4312 ) | = | A 2 n ( 1234 ) | .