  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.
