```Date: May 20, 2012 8:06 AM
Author: Vamph
Subject: Sets problem

Let S be the set: {1, 2, 3, ..., k}. How many sequences of k sets S_1,S_2,S_3, ...,S_k are there such that each set is a subset (not necessarily proper) of the next one,  and S_k is S?So, I am working on that question and could not figure out an easy way to solve it.Here is my attempt:The idea that I have is that we start from the set S_k and do its power set P(S_k). The sets included in P(S_k) are the different possible S_k-1. Then, for each of these possibilities we do the same things (the power sets of these sets to have the possible S_k-2) until we have all the possible sets for S_1. The number of these possible sets S1 will be the answer of that question. This is actually true, and can be tried for small values of k, but I think that it is impossible to apply it for k and compute the number of sequences. I also tried to see the problem from different perspectives, but none of them is applicable for k.I don't know if there is any way to know the total of the number of subsets of the subsets of a set. I think that it will solve the problem.
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