Wyvern has posted a walkthrough of the vault puzzle together with a map of the chains on
page one of the best tips. However, she suggests that one should go through all the chains pulling them - and then, once that is done, one should go back and pull those that have gone up. But if one pulls all the chains without going back all the chains will have gone up again.
This sequence works every time: 2, 1, 3, 4, 3, 5, 6, 2, 1, 5, 7, 4, 3. (Wyvern's numbers).