TY  - JOUR
AU  - Krityakierne, Tipaluck
AU  - Aek Thanatipanonda, Thotsaporn
PY  - 2025
TI  - A generating function framework for the no-feedback card guessing game after riffle shuffles
JO  - AIMS Mathematics
SP  - 24257
EP  - 24269
VL  - 10
IS  - 10
AB  - 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.
UR  - https://doi.org/10.3934/math.20251075
DO  - 10.3934/math.20251075