The total variation distance is an upper bound for the absolute difference in probability over all subsets of permutations.
In particular, we can consider the set:
U = {all permuations that have never been seen by a human}
The counting arguments in the article lead us to conclude that the uniform probability of U is very close to 1, i.e. almost all permuations have never been obtained by a human shuffle. The Bayer-Diaconis result implies that for certain types of shuffles the probability of ending up in U is at least
1 - (the total variation distance above)
So for 8 shuffles, we have a probablity of at least 82.4% of landing in U. This is considerably weaker than "every shuffle is unique".
In particular, we can consider the set:
The counting arguments in the article lead us to conclude that the uniform probability of U is very close to 1, i.e. almost all permuations have never been obtained by a human shuffle. The Bayer-Diaconis result implies that for certain types of shuffles the probability of ending up in U is at least So for 8 shuffles, we have a probablity of at least 82.4% of landing in U. This is considerably weaker than "every shuffle is unique".