On (shape-)Wilf-equivalence of certain sets of (partially ordered) patterns

TL;DR

This paper proves a conjecture on Wilf-equivalence of specific pattern sets, extending known results and establishing broad classes of pattern equivalences using generalized shape-Wilf-equivalence techniques.

Contribution

It introduces generalized shape-Wilf-equivalence results and proves a significant conjecture, expanding the understanding of pattern equivalences in combinatorics.

Findings

Proved the conjecture of Gao and Kitaev on Wilf-equivalence of two pattern sets.

Established shape-Wilf-equivalence for large classes of patterns.

Extended previous results to include 11 out of 12 classes related to the conjecture.

Abstract

We prove a conjecture of Gao and Kitaev on Wilf-equivalence of sets of patterns {12345,12354} and {45123,45213} that extends the list of 10 related conjectures proved in the literature in a series of papers. To achieve our goals, we prove generalized versions of shape-Wilf-equivalence results of Backelin, West, and Xin and use a particular result on shape-Wilf-equivalence of monotone patterns. We also derive general results on shape-Wilf-equivalence of certain classes of partially ordered patterns and use their specialization (also appearing in a paper by Bloom and Elizalde) as an essential piece in proving the conjecture. Our results allow us to show (shape-)Wilf-equivalence of large classes of sets of patterns, including 11 out of 12 classes found by Bean et al. in relation to the conjecture.