2023/09/16 by Chern, Shane, Jiu, Lin, Simonelli, Italo
#05A16 #60C05 #60F05 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2309.08841
Given a positive integer n, consider a random permutation τ of the set \1,2,…, n\. In τ, we look for sequences of consecutive integers that appear in adjacent positions: a maximal such a sequence is called a block. Each block in τ 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 Xn, the number of permutations needed for this process to end. In particular, we find an explicit asymptotic expression for each of E[Xn] and Var [Xn] as well as for every higher central moment, and show that Xn satisfies a central limit theorem.