Counting alternating permutations with restricted prefix and suffix

In this paper, we use Hasse diagrams and generating functions to count alternating permutations with restricted prefix and suffix of lengths 3 and 4. In other words, for an alternating permutation $\sigma ={\sigma }_1{\sigma }_2{\sigma }_3\cdots {\sigma }_n\in S_n$, we restrict length-3 prefixes ${\sigma }_1{\sigma }_2{\sigma }_3$ to follow certain patterns, such as $231$ and $132$, or follow certain restrictions such as ${\sigma }_2\ge \max \{{\sigma }_1,{\sigma }_3\}+2$, similarly for prefixes of length 4. We also study the enumeration of alternating permutations with restrictions on both prefix and suffix.