Information gain from discarded tiles in Mahjong
Like you said, you can't shuffle your tiles at will, but if I'm remembering correctly, you can put the tile you just drawn anywhere in your "hand", so you can't know when the discarded tile was drawn (at least not if you haven't a perfect memory ;) ).

However, each player has his own discard location, so you can easily see all tiles he has already discarded and maybe find a pattern in his discards which may prove useful to guess what he is waiting.

A player which is always discarding quickly the tile he just drawn is also a strong indicator that he is close to finish.