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 $0,1/n,2/n,\ldots,(n-1)/n,1$ and the multiplicity of the eigenvalue $i/n$ is equal to the number of the {\em 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.