Inverse Relations in Shapiro's Open Questions

As an inverse relation, involution with an invariant sequence plays a key role in combinatorics and features prominently in some of Shapiro's open questions [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585-596]. In this paper, invariant sequences are used to provide answers to some of these questions about the Fibonacci matrix and Riordan involutions.