2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69881We study generating functions for the number of involutions, even involutions, and odd involutions in $S_n$ subject to two restrictions. One restriction is that the involution avoid 3412 or contain 3412 exactly once. The other restriction is that the involution avoid another pattern $τ$ or contain $τ$ exactly once. In many cases we express these generating functions in terms of Chebyshev polynomials of the second kind.30 pagesCombinatorics05A05; 05A15; 30B70; 42C05text