A central limit theorem for a card shuffling problem

TL;DR

This paper studies a recursive permutation merging process, deriving its expected duration, variance, higher moments, and proving a central limit theorem for the number of permutations needed to reduce the set to one element.

Contribution

It provides explicit asymptotic formulas for the moments of the process duration and establishes a central limit theorem, advancing understanding of this permutation merging problem.

Findings

Asymptotic expressions for expected number of permutations

Explicit variance and higher moments formulas

Proof of a central limit theorem for the process

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.