  We are interested in the statistics of the length of the longest increasing
subsequence of 2-rowed lexicographically sorted arrays chosen according to
distinct families of distributions D = (D_n)_n, and when n goes to infinity.
This framework encompasses well studied problems such as the so called Longest
Increasing Subsequence problem, the Longest Common Subsequence problem,
problems concerning directed bond percolation models, among others. We define
several natural families of distinct distributions and characterize the
asymptotic behavior of the expected length of a longest increasing subsequence
chosen according to them. In particular, we consider generalizations to d-rowed
arrays as well as symmetry restricted two-rowed arrays.
