A generating function framework for the no-feedback card guessing game after riffle shuffles
Issued Date
2025-01-01
Resource Type
eISSN
24736988
Scopus ID
2-s2.0-105020056213
Journal Title
Aims Mathematics
Volume
10
Issue
10
Start Page
24257
End Page
24269
Rights Holder(s)
SCOPUS
Bibliographic Citation
Aims Mathematics Vol.10 No.10 (2025) , 24257-24269
Suggested Citation
Krityakierne T., Thanatipanonda T.A. A generating function framework for the no-feedback card guessing game after riffle shuffles. Aims Mathematics Vol.10 No.10 (2025) , 24257-24269. 24269. doi:10.3934/math.20251075 Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/112915
Title
A generating function framework for the no-feedback card guessing game after riffle shuffles
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Abstract
We introduce a generating-function framework for analyzing the no-feedback card-guessing game after k Gilbert–Shannon–Reeds riffle shuffles. We show that the distribution of the card appearing in position i can be expressed as a structured mixture of 2<sup>k</sup> tractable components, each corresponding to a sum of independent Bernoulli trials. From this decomposition, we derive an explicit closed-form expression for the probability generating function, represented as a product of binomial-type polynomials with a clear and systematic structure, valid for any number of cards n and any number of shuffles k. This formulation replaces recursive convolutions with a single analytic expression, enabling efficient computation and revealing the combinatorial–probabilistic structure underlying riffle shuffles. Beyond exact evaluation, the framework connects optimal no-feedback strategies with the generating functions and suggests asymptotic behavior in both the fixed-k, large-n and fixed-n, large-k regimes.