Permutations Involving 8 Elements

How many permutations \( \tau \) on 8 elements are there such that \( \tau \circ \tau \) is the identity permutation?

Details and assumptions

You may think of a permutation on 8 elements as a way to shuffle a deck of 8 cards. The identity permutation would correspond to the shuffle where we do nothing (hence it stays the same = identity). If \(\tau\) corresponds to only exchanging the top 2 cards, then \( \tau \circ \tau\) would return the deck to it's unchanged state.

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