On Longest Increasing Subsequences and Random Young Tableaux: Experimental Results and Recent Theorems

Ulam (1961) apparently first posed the following question: what is the average (or distribution of) the length Ln of the longest increasing subsequence of a random permutation of the first n integers? Experimental (Monte-Carlo) evidence has played an important role in the study of Ln begining with Ulam (1961), and continuing with Baer and Brock (1969), and Odlyzko and Rains (2000). We present experimental evidence concerning the distribution of the length Ln of the longest increasing increasing subsequence of a random permutation of length n. In particular, the experimental data confirm the known result E(Ln) ∼ 2 √ n and strongly suggest that V ar(Ln) ∼ cn for a constant c ≈ .818 . . .. This supports and complements the recent results of Baik, Deift, and Johansson (1999), who apparently knew of the monte-carlo results of Odlyzko and Rains (2000). In the last section we also combine our experimental results with those of Odlyzko and Rains (2000). AMS 2000 subject classifications. Primary: 60G15, 60G99; secondary 60E05.