Folding Phenomenon of Major-balance Identities on Restricted Involutions

TL;DR

This paper proves refined major-balance identities for specific classes of involutions avoiding certain patterns, using combinatorial involutions and lattice path techniques, advancing understanding of permutation pattern avoidance.

Contribution

It introduces new refined identities for 321- and 123-avoiding involutions, employing sign-reversing involutions and lattice path analysis.

Findings

Refined major-balance identity for 321-avoiding involutions respecting leading element.

Affirmative proof of a refined identity for 123-avoiding involutions respecting descents.

Use of combinatorial involutions and lattice paths in proving permutation identities.

Abstract

In this paper we prove a refined major-balance identity on the $321$-avoiding involutions of length $n$, respecting the leading element of permutations. The proof is based on a sign-reversing involution on the lattice paths within a $\lfloor\frac{n}{2}\rfloor\times\lceil\frac{n}{2}\rceil$ rectangle. Moreover, we prove affirmatively a question about refined major-balance identity on the $123$-avoiding involutions, respecting the number of descents.