A New Class of Wilf-Equivalent Permutations

TL;DR

This paper introduces a new class of Wilf-equivalent permutation patterns, completing the classification for patterns of length 7 and resolving previously missing equivalences.

Contribution

It establishes a novel class of Wilf-equivalent permutations, expanding the understanding of permutation pattern equivalences for lengths up to 7.

Findings

Identifies a new class of Wilf-equivalent patterns involving (n-1,n-2,n,tau) and (n-2,n,n-1,tau)

Completes the classification of permutation patterns for length 7

Resolves the last remaining cases in S_7

Abstract

For about 10 years, the classification of permutation patterns was thought completed up to length 6. In this paper, we establish a new class of Wilf-equivalent permutation patterns, namely, (n-1,n-2,n,tau)~(n-2,n,n-1,tau) for any tau in S_{n-3}. In particular, at level n=6, this result includes the only missing equivalence (546213)~(465213), and for n=7 it completes the classification of permutation patterns by settling all remaining cases in S_7.