Pattern avoidance by even permutations

TL;DR

This paper investigates even-Wilf-equivalence in pattern avoidance by permutations in the alternating group, establishing which classical Wilf-equivalence results hold under this restriction and classifying patterns of certain lengths.

Contribution

It extends Wilf-equivalence concepts to the even permutation setting, proving new equivalences and classifying patterns of lengths 4, 5, and 6.

Findings

Certain Wilf-equivalence results hold for even permutations

Classification of patterns of length 4, 5, and 6 under even-Wilf-equivalence

Some classical Wilf-equivalence results do not extend to even permutations

Abstract

We study questions of even-Wilf-equivalence, the analogue of Wilf-equivalence when attention is restricted to pattern avoidance by permutations in the alternating group. Although some Wilf-equivalence results break when considering even-Wilf-equivalence analogues, we prove that other Wilf-equivalence results continue to hold in the even-Wilf-equivalence setting. In particular, we prove that t(t-1)...321 and (t-1)(t-2)...21t are even-shape-Wilf-equivalent for odd t, paralleling a result (which held for all t) of Backelin, West, and Xin for shape-Wilf-equivalence. This allows us to classify the patterns of length 4, and to partially classify patterns of length 5 and 6. As with transition to involution-Wilf-equivalence, some (but not all) of the classical Wilf-equivalence results are preserved when we make the transition to even-Wilf-equivalence.