Asymptotic normality and combinatorial aspects of the prefix exchange distance distribution

TL;DR

This paper studies the distribution of prefix exchange distances in permutations, providing new combinatorial proofs, explicit formulas for mean and variance, and demonstrating asymptotic normality.

Contribution

It offers novel combinatorial proofs and explicit formulas for the mean and variance, along with establishing the normality of the distribution's asymptotic behavior.

Findings

Distribution of prefix exchange distance is asymptotically normal.

Explicit formulas for mean and variance of the distribution.

New combinatorial proofs of known results.

Abstract

The prefix exchange distance of a permutation is the minimum number of exchanges involving the leftmost element that sorts the permutation. We give new combinatorial proofs of known results on the distribution of the prefix exchange distance for a random uniform permutation. We also obtain expressions for the mean and the variance of this distribution, and finally, we show that the normalised prefix exchange distribution converges in distribution to the standard normal distribution.