Limit Theorems for Descents and Inversions of Shelf-Shuffles
Alexander Clay
TL;DR
This work analyzes permutation statistics for shelf-shuffles, proving central limit theorems for the number of inversions and descents as the deck size $n$ grows. It leverages a random-word decomposition to express inversions and derives the mean $E I_{n,m}=\frac{n(n-1)}{4}$ and variance $Var(I_{n,m})=\frac{n(n-1)(2m^2 n+4n+5m^2+18m-17)}{72m^2}$; a $U$-statistic framework then yields a CLT for inversions with a rate $d_K(\tilde{I}_{n,m},Z) \le C/\sqrt{n}$ that is independent of the number of shelves $m$. For descents, the authors couple $d_{n,m}$ to a $Binomial(n,1/2)$ variable with an almost-sure bound $|d_{n,m}-B_{n,m}|\le 4m-1$, which, under $m=o(\sqrt{n})$, gives a CLT with asymptotic variance $(m^2+2)/(3m^2)$. The paper also discusses extending these results to biased shelf-shuffles, linking shelf-shuffle statistics to random-word models and potential applications to related partition structures.
Abstract
We prove central limit theorems for the number of descents and the number of inversions after a shelf-shuffle. In particular, we bound the convergence rate for the number of inversions independently of the number of shelves. Along the way, we determine the mean and variance for the number of inversions after a shelf shuffle, which was also an open problem. We also suggest ways to extend our results to biased shelf-shuffles.