New equivalences for pattern avoiding involutions

TL;DR

This paper completes the classification of pattern avoidance in signed involutions and permutations of lengths 5 to 7, proving conjectures and establishing new equivalences among patterns.

Contribution

It provides the full Wilf classification for signed and unsigned permutations and involutions of lengths 5 to 7, including new pattern equivalences and proofs of conjectures.

Findings

Complete Wilf classification for S_5, S_6, S_7

New pattern equivalences for involutions

Proof of Jaggard's conjectures

Abstract

We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S_5, S_6, and S_7 for both permutations and involutions.