We compute the limiting distributions of the lengths of the longest
monotone subsequences of random (signed) involutions with or without
conditions on the number of fixed points (and negated points) as the
sizes of the involutions tend to infinity. The resulting distributions
are, depending on the number of fixed points, (1) the Tracy-Widom
distributions for the largest eigenvalues of random GOE, GUE, GSE
matrices, (2) the normal distribution, or (3) new classes of
distributions which interpolate between pairs of the Tracy-Widom
distributions. We also consider the second rows of the corresponding
Young diagrams. In each case the convergence of moments is also
shown. The proof is based on the algebraic work of J. Baik and
E. Rains in [7] which establishes a connection between the statistics
of random involutions and a family of orthogonal polynomials, and an
asymptotic analysis of the orthogonal polynomials which is obtained by
extending the Riemann-Hilbert analysis for the orthogonal polynomials
by P. Deift, K. Johansson, and Baik in [3].