Folding Phenomenon of Major-balance Identities on Restricted Involutions

Abstract

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