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

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.

💡 Research Summary

The paper provides a complete spectral analysis of the position matrix M associated with a single‑shelf shuffle, a shuffling scheme used in many casino mechanical shufflers. In a single‑shelf shuffle each incoming card is independently placed on the top or bottom of a single shelf with probability ½, and after one round the probability that the i‑th incoming card ends up in position j is
(M(i,j)=\frac12\binom{i-1}{j-1}+\frac12\binom{i-1}{n-j}) for 1≤i,j≤n.
The authors factor M as (M=L(I+P)) where L is a lower‑triangular matrix with diagonal entries (2^{-i}) and P is the reversal permutation matrix. They introduce the falling‑factorial basis (v^{(j)}_k=(k-1)^j) (j=0,…,n‑1). In this basis L is diagonal with eigenvalues (2^{-j-1}) and (I+P) becomes upper‑triangular. Consequently M is upper‑triangular in the same basis, and its eigenvalues are exactly (2^{-i}) for even i (0≤i≤n‑1) and 0 for odd i. The corresponding eigenvectors are given explicitly by a linear combination of falling factorials with coefficients involving Bernoulli numbers: \