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On the statistics of random-to-top shuffles

Alexander Clay

Abstract

We prove limit theorems for the number of fixed points, descents, and inversions of iterated random-to-top shuffles in two limiting cases. Our proofs are analytical and rely on novel combinatorial decompositions of each statistic into randomly indexed statistics of uniformly random permutations. New combinatorial proofs of the expected number of fixed points and inversions are given. Our results answer questions of Diaconis, Fulman, and Pehlivan.

On the statistics of random-to-top shuffles

Abstract

We prove limit theorems for the number of fixed points, descents, and inversions of iterated random-to-top shuffles in two limiting cases. Our proofs are analytical and rely on novel combinatorial decompositions of each statistic into randomly indexed statistics of uniformly random permutations. New combinatorial proofs of the expected number of fixed points and inversions are given. Our results answer questions of Diaconis, Fulman, and Pehlivan.
Paper Structure (12 sections, 23 theorems, 91 equations, 3 figures)

This paper contains 12 sections, 23 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Outline
  5. Notation
  6. Permutation Statistics
  7. Balls in Boxes
  8. Combinatorics of Random-to-Top Shuffles
  9. Fixed Points
  10. Descents
  11. Inversions
  12. Future Directions

Key Result

Theorem 1.1

Let $F^r_n$ be the number of fixed points after $r$ iterated random-to-top shuffles of an $n$-card deck, starting from the identity. Let $c>0$ be a constant. We have where $X\sim\operatorname{Poisson}(1-e^{-c})$ and $Y$ is geometric with parameter $1-e^{-c}$ and $P(Y=k)=(1-e^{-c})(e^{-ck})$ for all $k\geq 0$. Moreover, $X$ and $Y$ are independent. If $r\gg n$, then

Figures (3)

  • Figure 1: Histograms of $2000$ trials of the number of fixed points of $n=10000$ card decks after iterated RTT shuffles, normalized to be a probability distribution. The overlays are linearly interpolated versions of the predicted Poisson-geometric convolutions from Theorem \ref{['fixed points main']} in blue and a $\operatorname{Poisson}(1)$ distribution in orange. The top histogram represents $r=5000$ shuffles, the middle $r=10000$ shuffles, the bottom $r=20000$ shuffles. The decks given $5000$, respectively $10000$ shuffles are "retired".
  • Figure 2: Histograms of $2000$ trials of the number of descents of $n=10000$ card decks after iterated RTT shuffles centered by the mean and scaled by $n^{-1/2}$, normalized to be a probability distribution. Top: $r=2500$ shuffles, middle $r=5000$, bottom $r=10000$. The cyan curves are the predicted $r=cn$ densities from Theorem \ref{['descents main']}. The orange curves are $\mathcal{N}(0,1/12)$ densities. The decks given $2500$ and $5000$ shuffles are all "retired".
  • Figure 3: Histograms of $1000$ trials of the number of inversions of $n=1000$ card decks after iterated RTT shuffles centered by the mean and scaled by $n^{-3/2}$, normalized to be a probability distribution. Top: $r=100$ shuffles, middle $r=250$, bottom $r=1000$. The cyan curves are the predicted $r=cn$ densities from Theorem \ref{['inversions main']}. The orange curves are $\mathcal{N}(0,1/36)$ densities. The decks given $100$ and $250$ shuffles are all "retired".

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Theorem 2.2: Slutsky
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Theorem 4.1
  • ...and 34 more