Values of L_n, the length of the longest increasing subsequences in 
10^7 random permutations on {1, 2, ..., n} for n = 10^4.  There were 
3 permutations in which L_n was 175, 9 in which it was 176, and so on.

   3 175
   9 176
  66 177
 292 178
1076 179
3311 180
9290 181
23382 182
51671 183
102644 184
184339 185
300683 186
447326 187
610765 188
767631 189
896422 190
976770 191
993397 192
951982 193
860347 194
736345 195
596794 196
462063 197
342126 198
242972 199
165447 200
108258 201
68506 202
41636 203
24286 204
13940 205
7815 206
4158 207
2185 208
1075 209
 536 210
 252 211
 106 212
  51 213
  25 214
  12 215
   4 216
   2 217