In class I introduced the -analog of and :

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Let be a power of a prime and let be the field of elements. In several senses, the -analog of the set is the vector space . Here is one sense:

1. Prove that the number of sequences of sets is .

2. Prove that the number of sequences of subspaces is .

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1. The number of sequences of sets is equal to the number of permutations of , consider the following bijection: for a permutation you can get the sequence given by (). Therefore, the number of such sequences is

2. Note that given a subspace in the sequence , to get we only need to add any nonzero vector in and take the generated space of the union. has dimension and therefore nonzero vectors, and two of these give the same subspace if and only if they generate the same space, i.e. if and only if one is a nonzero-multiple of the other. So given , there are possibilities for .

Then, there are possibilities for , for , …, for and for . So the number of sequences is:

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