Although I am by no means an applied mathematician, I like to occasionally explain applications of the math I teach to real world problems. Right now I am teaching some students about longest increasing subsequences of permutations and their connection to the Robinson-Schensted algorithm.
Suppose you have some discrete time series data $x_1,x_2,\ldots$. This determines a permutation $\sigma = \sigma_1, \sigma_2,\ldots$ just by relative order of the data (if the data is continuous, or at least drawn from a wide range, it is very likely there will be no "ties" so we have an honest permutation). I remember once hearing (but don't remember where...) that checking the length of the longest increasing subsequence of $\sigma$ could be a way to detect if the data was observed in a random order or not.
But a cursory Googling (which gives a lot of info about longest increasing subsequences, connections to RS, "Ulam's problem" of computing the expected length, dynamic programming for finding a l.i.s., etc.) does not yield much about this application...
Question: Is comparing the length of the longest increasing subsequence of $\sigma$ to that of a random permutation a reasonable statistical test for randomness? Is there some literature about this?
Of course, for a more sophisticated test we could look at the shape of $\sigma$ under RS and compare it to the known Vershik–Kerov/Logan–Shepp limit shape.