A central limit theorem for a card shuffling problem

Abstract

Given a positive integer $n$, consider a random permutation $\tau$ of the set $\{1,2,\ldots, n\}$. In $\tau$, we look for sequences of consecutive integers that appear in adjacent positions: a maximal such a sequence is called a block. Each block in $\tau$ is merged, and after all the merges, the elements of this new set are relabeled from $1$ to the current number of elements. We continue to randomly permute and merge this new set until only one integer is left. In this paper, we investigate the asymptotic behavior of $X_n$, the number of permutations needed for this process to end. In particular, we find an explicit asymptotic expression for each of $\mathbf{E}[X_n]$ and $\mathbf{Var} [X_n]$ as well as for every higher central moment, and show that $X_n$ satisfies a central limit theorem.