Some statistics on restricted 132 involutions

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.