Top to random shuffles on colored permutations

Abstract

A deck of n cards are shuffled by repeatedly taking off the top card, flipping it with probability 1/2, and inserting it back into the deck at a random position. This process can be considered as a Markov chain on the group B n of signed permutations. We show that the eigenvalues of the transition probability matrix are 0, 1/n, 2/n, . . . , (n − 1)/n, 1 and the multiplicity of the eigenvalue i/n is equal to the number of the signed permutation having exactly i fixed points. We show the similar results also for the colored permutations. Further, we show that the mixing time of this Markov chain is n log n, same as the ordinary 'top-to-random' shuffles without flipping the cards. The cut-off is also analyzed by using the asymptotic behavior of the Stirling numbers of the second kind.