  An alternating permutation of length $n$ is a permutation $\pi=\pi_1 \pi_2
... \pi_n$ such that $\pi_1 < \pi_2 > \pi_3 < \pi_4 > ...$. Let $A_n$ denote
set of alternating permutations of ${1,2,..., n}$, and let $A_n(\sigma)$ be set
of alternating permutations in $A_n$ that avoid a pattern $\sigma$. 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 B\'ona, 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.
