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On the position matrix of single-shelf shuffle and card guessing

Raghavendra Tripathi

TL;DR

This work resolves the spectral structure of the position matrix for a single-shelf shuffle by constructing explicit eigenvectors and a full diagonalization using a falling factorial basis and Bernoulli numbers. The authors show that the spectrum consists of 0 together with nonzero eigenvalues 2^{-i} for even i and provide closed-form left and right eigenvectors, enabling a complete spectral decomposition M = XDX^{-1}. These results yield sharp consequences for card guessing without feedback: after k shuffles with k exceeding (1+ε) log n, the maximum expected number of correct guesses is 1 plus a vanishing term, and for a single shuffle the no-feedback score is on the order of sqrt(2n/pi) with an explicit simple strategy attaining sqrt(2n/pi) minus 1. The diagonalization also clarifies that the no-feedback conjectured strategies are not optimal and provides concrete, provable bounds for general k. Overall, the paper connects spectral data to guessing performance and resolves prior conjectures while offering precise asymptotics relevant to practical shelf-shuffling contexts.

Abstract

Mechanical shufflers used in many casinos employ a card shuffling scheme called \emph{shelf shuffling}. In a single-shelf shuffling, cards arrive sequentially, and each incoming card is independently placed on the top or the bottom of a shelf with equal probability. The position matrix of a single-shelf shuffling encodes the probability that the $i$-th incoming card is in position $j$ after one round of single-shelf shuffle. The spectral properties of the position matrix of card shuffling schemes are helpful in the analysis of card guessing games without feedback. In this paper, we determine the full spectrum and the corresponding eigenspaces of the position matrix $M$ of a single-shelf shuffle. This strengthens and resolves two conjectures in a recent work [arXiv:2507.10294]. As a consequence of these results, we show that the maximum number of expected correct guesses without feedback after $k\geq (1+ε)$ many shuffles is of the order $1+O(n^{-2ε})$. On the other hand, the expected number of correct guesses after one shuffle is at most $\sqrt{2n/π}+1+O(n^{-1/2})$, and we give a strategy (not optimal) that achieves $\sqrt{2n/π}-1$ number of correct cards in expectation.

On the position matrix of single-shelf shuffle and card guessing

TL;DR

This work resolves the spectral structure of the position matrix for a single-shelf shuffle by constructing explicit eigenvectors and a full diagonalization using a falling factorial basis and Bernoulli numbers. The authors show that the spectrum consists of 0 together with nonzero eigenvalues 2^{-i} for even i and provide closed-form left and right eigenvectors, enabling a complete spectral decomposition M = XDX^{-1}. These results yield sharp consequences for card guessing without feedback: after k shuffles with k exceeding (1+ε) log n, the maximum expected number of correct guesses is 1 plus a vanishing term, and for a single shuffle the no-feedback score is on the order of sqrt(2n/pi) with an explicit simple strategy attaining sqrt(2n/pi) minus 1. The diagonalization also clarifies that the no-feedback conjectured strategies are not optimal and provides concrete, provable bounds for general k. Overall, the paper connects spectral data to guessing performance and resolves prior conjectures while offering precise asymptotics relevant to practical shelf-shuffling contexts.

Abstract

Mechanical shufflers used in many casinos employ a card shuffling scheme called \emph{shelf shuffling}. In a single-shelf shuffling, cards arrive sequentially, and each incoming card is independently placed on the top or the bottom of a shelf with equal probability. The position matrix of a single-shelf shuffling encodes the probability that the -th incoming card is in position after one round of single-shelf shuffle. The spectral properties of the position matrix of card shuffling schemes are helpful in the analysis of card guessing games without feedback. In this paper, we determine the full spectrum and the corresponding eigenspaces of the position matrix of a single-shelf shuffle. This strengthens and resolves two conjectures in a recent work [arXiv:2507.10294]. As a consequence of these results, we show that the maximum number of expected correct guesses without feedback after many shuffles is of the order . On the other hand, the expected number of correct guesses after one shuffle is at most , and we give a strategy (not optimal) that achieves number of correct cards in expectation.
Paper Structure (10 sections, 17 theorems, 94 equations)

This paper contains 10 sections, 17 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Contributions of this paper
  3. Results
  4. Proof of Theorem \ref{['thm:Main Theorem1']}:
  5. An eigenbasis of L
  6. I plus P is upper triangular in the basis B
  7. Non-trivial eigenvectors of T
  8. Proof of Theorem \ref{['thm: Ciucu']}
  9. Diagonalization of M
  10. Enrtywise bounds on zeta(i) and tilde zeta(i)

Key Result

Theorem I

Fix $n\geq 2$ and let $M$ be the position matrix as defined above. Then,

Theorems & Definitions (34)

  • Definition 1.1: $m$-Shelf shuffle
  • Conjecture 1.2
  • Theorem I
  • Remark 1.3
  • Theorem II
  • Remark 1.4
  • Theorem III
  • proof : Proof of Theorem \ref{['thm:Main Theorem1']}(1)
  • Proposition 2.1
  • proof
  • ...and 24 more