  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.
