Counting alternating permutations with restricted prefix and suffix

TL;DR

This paper develops methods using Hasse diagrams and generating functions to count alternating permutations with specific prefix and suffix restrictions, advancing combinatorial enumeration techniques.

Contribution

It introduces new enumeration formulas for restricted alternating permutations, including patterns and inequalities on prefixes and suffixes.

Findings

Derived explicit counting formulas for restricted permutations

Analyzed permutations with combined prefix and suffix restrictions

Extended enumeration techniques to new classes of permutations

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 $\sigma=\sigma_1\sigma_2\sigma_3\cdots\sigma_{n}\in S_{n}$, 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 $\sigma_2 \geq \max\{\sigma_1,\sigma_3\}+2$, similarly for prefixes of length 4. We also study the enumeration of alternating permutations with restrictions on both prefix and suffix.