Values of L_n, the length of the longest increasing subsequences in 
6*10^5 random permutations on {1, 2, ..., n} for n = 10^5.  There were 
3 permutations in which L_n was 598, 8 in which it was 599, and so on.

   3 598
   8 599
  16 600
  45 601
  96 602
 188 603
 399 604
 710 605
1277 606
2073 607
3295 608
5075 609
7370 610
10311 611
14095 612
17941 613
22252 614
26762 615
30912 616
34511 617
37247 618
39588 619
39939 620
39285 621
38108 622
35792 623
33018 624
29514 625
25865 626
21892 627
18237 628
15057 629
12108 630
9533 631
7446 632
5638 633
4268 634
2977 635
2213 636
1592 637
1169 638
 720 639
 503 640
 347 641
 219 642
 143 643
  92 644
  47 645
  42 646
  28 647
  13 648
   5 649
   8 650
   5 651
   1 652
   1 653
   1 657