Reversing the deck after a top-to-random shuffle gives an operator whose minimal polynomial is a product of distinct linear factors, with roots the signed knapsack numbers: for the basic reverse shuffle, the integers -n+2, 0, n, and every integer from -n+4 to n-3.
Semigroup and ring theoretical methods in probability
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Top to random and reverse: analysis of a new descent algebra shuffle
Reversing the deck after a top-to-random shuffle gives an operator whose minimal polynomial is a product of distinct linear factors, with roots the signed knapsack numbers: for the basic reverse shuffle, the integers -n+2, 0, n, and every integer from -n+4 to n-3.