@article{Krityakierne2025, 
author = {Tipaluck Krityakierne and Thotsaporn Aek Thanatipanonda},
title = {A generating function framework for the no-feedback card guessing game after riffle shuffles},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {10},
pages = {24257-24269},
keywords = {card guessing, no-feedback strategy, Gilbert–Shannon–Reeds riffle shuffle, probability generating function, mixture distribution},
url = {https://www.sciopen.com/article/10.3934/math.20251075},
doi = {10.3934/math.20251075},
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    k   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.}
}