  In [GM] Guibert and Mansour studied involutions on n letters avoiding (or
containing exactly once) 132 and avoiding (or containing exactly once) an
arbitrary pattern on k letters. They also established a bijection between
132-avoiding involutions and Dyck word prefixes of same length. Extending this
bijection to bilateral words allows to determine more parameters; in
particular, we consider the number of inversions and rises of the involutions
onto the words. This is the starting point for considering two different
directions: even/odd involutions and statistics of some generalized patterns.
Thus we first study generating functions for the number of even or odd
involutions on n letters avoiding (or containing exactly once) 132 and avoiding
(or containing exactly once) an arbitrary pattern $\tau$ on k letters. In
several interesting cases the generating function depends only on k and is
expressed via Chebyshev polynomials of the second kind. Next, we consider other
statistics on 132-avoiding involutions by counting an occurrences of some
generalized patterns, related to the enumeration according to the number of
rises.
