Folding Phenomenon of Major-balance Identities on Restricted Involutions

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.