Families of Shape-Wilf-Equivalent Claw-Shaped Partially Ordered Patterns

TL;DR

This paper extends the understanding of shape-Wilf-equivalence for claw-shaped partially ordered patterns (POPs) by establishing equivalences for families of such patterns and introducing a novel encoding method.

Contribution

It introduces a new encoding approach to prove shape-Wilf-equivalence for families of claw-shaped POPs, expanding prior results.

Findings

Established shape-Wilf-equivalence for certain families of claw-shaped POPs

Provided enumeration formulas for permutations avoiding these POPs

Developed a novel encoding method different from previous approaches

Abstract

Partially ordered patterns (POPs) generalize classical permutation patterns and have been extensively studied in the contexts of permutations, words, compositions, and partitions. Burstein, Han, Kitaev, and Zhang established the shape-Wilf-equivalence for individual claw-shaped POPs. In this paper, we extend their result by proving that certain families of claw-shaped POPs are shape-Wilf-equivalent and enumerate the number of permutations avoiding that set of claw-shaped POPs. Our approach is based on a new encoding process, which is entirely different from the method used in their work.