In this paper we prove refined major-balance identities on the 321-avoiding involutions of length ??, respecting the leading element of permutations. The proof is based on sign-reversing involutions on the lattice paths within a [??/2] × [??/2] rectangle. Moreover, we prove afffirmatively a question about refined major-balance identities on the 123-avoiding involutions, respecting the number of descents.
Taiwanese Journal of Mathematics 22(5), p.1031-1050