Suppose I have a deck of cards in some ordering. Consider all of the other possible orderings of this same deck relative to the first one. Is there a bound we can put on the maximum number of inversions needed to get from our original deck to any other shuffled ordering.
i.e. In a deck of 52 cards ($c_1, c_2,..., c_{52}$), we have an upper bound (but not necessarily the least?) of 51 inversions. The method would look like placing the shuffled $c_{1}$ at the front. Then placing the $c_2$. etc..
So I suppose the question becomes, is there a sorting algorithm that only performs inversions that guarantees a max number of inversions less than 51 for a deck of size 52?