Counting alternating permutations with restricted prefix and suffix

Abstract

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 $\ddot{I}ƒ=\ddot{I}{ƒ}_1\ddot{I}{ƒ}_2\ddot{I}{ƒ}_3\cdots \ddot{I}{ƒ}_n\in S_n$, we restrict length-3 prefixes $\ddot{I}{ƒ}_1\ddot{I}{ƒ}_2\ddot{I}{ƒ}_3$ to follow certain patterns, such as $231$ and $132$, or follow certain restrictions such as $\ddot{I}{ƒ}_2\ge \max \{\ddot{I}{ƒ}_1,\ddot{I}{ƒ}_3\}+2$, similarly for prefixes of length 4. We also study the enumeration of alternating permutations with restrictions on both prefix and suffix.