Markov chains arising from biased random derangements

We explore the cycle types of a class of biased random derangements, described as a random game played by some children labeled $1,\cdots,n$. Children join the game one by one, in a random order, and randomly form some circles of size at least $2$, so that no child is left alone. The game gives rise to the cyclic decomposition of a random derangement, inducing an exchangeable random partition. The rate at which the circles are closed varies in time, and at each time $t$, depends on the number of individuals who have not played until t. A $\{0,1\}$-valued Markov chain $X^n$ records the cycle type of the corresponding random derangement in that any $1$ represents a hand-grasping event that closes a circle. Using this, we study the cycle counts and sizes of the random derangements and their asymptotic behavior. We approximate the total variation distance between the reversed chain of $X^n$ and its weak limit $X^\infty$, as $n\to\infty$. We establish conditional (and push-forward) relations between $X^n$ and a generalization of the Feller coupling, given that no $11$-pattern ($1$-cycle) appears in the latter. We extend these relations to $X^\infty$ and apply them to investigate some asymptotic behaviors of $X^n$.