Permutations avoiding 312 and another pattern, Chebyshev polynomials and longest increasing subsequences

We study the longest increasing subsequence problem for random permutations avoiding the pattern 312 and another pattern τ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutation classes specifically when the pattern τ is monotone increasing or decreasing, or any pattern of length four.