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  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 .
Jinho Baik. Eric M. Rains. "The asymptotics of monotone subsequences of involutions." Duke Math. J. 109 (2) 205 - 281, 15 Ausust 2001. https://doi.org/10.1215/S0012-7094-01-10921-6