Two refined major-balance identities on 321-avoiding involutions

Making use of a combinatorial approach, we prove two refined major-balance identities on the 321-avoiding involutions in Sn, respecting the number of fixed points and the number of descents, respectively. The former one is proved in terms of ordered trees whose non-root nodes have exactly two children, and the latter one is proved in terms of lattice paths within a ⌊n2⌋×⌈n2⌉ rectangle.